sticksstones12 - Mathbox for metakunt

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Theorem sticksstones12 42643

Description: Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)

**Hypotheses**
Ref	Expression
sticksstones12.1	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
sticksstones12.2	⊢ (𝜑 → 𝐾 ∈ ℕ)
sticksstones12.3	⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))
sticksstones12.4	⊢ 𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))
sticksstones12.5	⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}
sticksstones12.6	⊢ 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}

**Assertion**
Ref	Expression
sticksstones12	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Distinct variable groups: 𝐴,𝑎,𝑗,𝑘,𝑙 𝐴,𝑏,𝑘,𝑥,𝑦 𝑥,𝑎,𝑦,𝑗,𝑘,𝑙 𝐵,𝑎,𝑖,𝑘,𝑙 𝐵,𝑏 𝐵,𝑗 𝐹,𝑏,𝑘 𝐾,𝑎,𝑓,𝑗,𝑙,𝑥,𝑦 𝐾,𝑏,𝑘,𝑔,𝑖 𝑔,𝑎,𝑖,𝑘,𝑁 𝑓,𝑁,𝑗,𝑙 𝑁,𝑏,𝑘,𝑔,𝑖 𝜑,𝑎,𝑖,𝑘,𝑙 𝜑,𝑏 𝜑,𝑗,𝑥,𝑦 𝑖,𝑏 𝑓,𝑏,𝑥,𝑦 𝑔,𝑏 Allowed substitution hints: 𝜑(𝑓,𝑔) 𝐴(𝑓,𝑔,𝑖) 𝐵(𝑥,𝑦,𝑓,𝑔) 𝐹(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑎,𝑙) 𝐺(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑎,𝑏,𝑙) 𝑁(𝑥,𝑦)

Proof of Theorem sticksstones12 Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sticksstones12.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
2		sticksstones12.2	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ)
3	2	nnnn0d 12489	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
4		sticksstones12.3	. . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))
5		sticksstones12.5	. . 3 ⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}
6		sticksstones12.6	. . 3 ⊢ 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}
7	1, 3, 4, 5, 6	sticksstones8 42638	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
8		sticksstones12.4	. . 3 ⊢ 𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))
9	1, 2, 8, 5, 6	sticksstones10 42640	. 2 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
10	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1)))))))
11		0red 11138	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℝ)
12	2	nngt0d 12217	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 𝐾)
13	11, 12	ltned 11273	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≠ 𝐾)
14	13	necomd 2989	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ≠ 0)
15	14	neneqd 2939	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝐾 = 0)
16	15	iffalsed 4465	. . . . . . . . 9 ⊢ (𝜑 → if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))) = (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1)))))
17	16	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))) = (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1)))))
18	17	mpteq2dva 5165	. . . . . . 7 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1)))))) = (𝑏 ∈ 𝐵 ↦ (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))
19	10, 18	eqtrd 2774	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))
20	19	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))
21		fveq1 6826	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑐) → (𝑏‘𝐾) = ((𝐹‘𝑐)‘𝐾))
22	21	oveq2d 7372	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑐) → ((𝑁 + 𝐾) − (𝑏‘𝐾)) = ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)))
23		fveq1 6826	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐹‘𝑐) → (𝑏‘1) = ((𝐹‘𝑐)‘1))
24	23	oveq1d 7371	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑐) → ((𝑏‘1) − 1) = (((𝐹‘𝑐)‘1) − 1))
25		fveq1 6826	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝐹‘𝑐) → (𝑏‘𝑘) = ((𝐹‘𝑐)‘𝑘))
26		fveq1 6826	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝐹‘𝑐) → (𝑏‘(𝑘 − 1)) = ((𝐹‘𝑐)‘(𝑘 − 1)))
27	25, 26	oveq12d 7374	. . . . . . . . . . 11 ⊢ (𝑏 = (𝐹‘𝑐) → ((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) = (((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))))
28	27	oveq1d 7371	. . . . . . . . . 10 ⊢ (𝑏 = (𝐹‘𝑐) → (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1) = ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1))
29	24, 28	ifeq12d 4476	. . . . . . . . 9 ⊢ (𝑏 = (𝐹‘𝑐) → if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1)) = if(𝑘 = 1, (((𝐹‘𝑐)‘1) − 1), ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1)))
30	22, 29	ifeq12d 4476	. . . . . . . 8 ⊢ (𝑏 = (𝐹‘𝑐) → if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))) = if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)), if(𝑘 = 1, (((𝐹‘𝑐)‘1) − 1), ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1))))
31	30	adantr 481	. . . . . . 7 ⊢ ((𝑏 = (𝐹‘𝑐) ∧ 𝑘 ∈ (1...(𝐾 + 1))) → if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))) = if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)), if(𝑘 = 1, (((𝐹‘𝑐)‘1) − 1), ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1))))
32	31	mpteq2dva 5165	. . . . . 6 ⊢ (𝑏 = (𝐹‘𝑐) → (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1)))) = (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)), if(𝑘 = 1, (((𝐹‘𝑐)‘1) − 1), ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1)))))
33	32	adantl 482	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑏 = (𝐹‘𝑐)) → (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1)))) = (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)), if(𝑘 = 1, (((𝐹‘𝑐)‘1) − 1), ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1)))))
34	7	ffvelcdmda 7025	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐹‘𝑐) ∈ 𝐵)
35		fzfid 13926	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (1...(𝐾 + 1)) ∈ Fin)
36	35	mptexd 7168	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)), if(𝑘 = 1, (((𝐹‘𝑐)‘1) − 1), ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1)))) ∈ V)
37	20, 33, 34, 36	fvmptd 6943	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐺‘(𝐹‘𝑐)) = (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)), if(𝑘 = 1, (((𝐹‘𝑐)‘1) − 1), ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1)))))
38	4	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))))
39		simpllr 781	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑎 = 𝑐) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑙 ∈ (1...𝑗)) → 𝑎 = 𝑐)
40	39	fveq1d 6829	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑎 = 𝑐) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑙 ∈ (1...𝑗)) → (𝑎‘𝑙) = (𝑐‘𝑙))
41	40	sumeq2dv 15655	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑎 = 𝑐) ∧ 𝑗 ∈ (1...𝐾)) → Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙) = Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙))
42	41	oveq2d 7372	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑎 = 𝑐) ∧ 𝑗 ∈ (1...𝐾)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)) = (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))
43	42	mpteq2dva 5165	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑎 = 𝑐) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙))))
44		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ 𝐴)
45		fzfid 13926	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (1...𝐾) ∈ Fin)
46	45	mptexd 7168	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙))) ∈ V)
47	38, 43, 44, 46	fvmptd 6943	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐹‘𝑐) = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙))))
48		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑗 = 𝐾) → 𝑗 = 𝐾)
49	48	oveq2d 7372	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑗 = 𝐾) → (1...𝑗) = (1...𝐾))
50	49	sumeq1d 15653	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑗 = 𝐾) → Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙) = Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙))
51	48, 50	oveq12d 7374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑗 = 𝐾) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)) = (𝐾 + Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙)))
52		1zzd 12549	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℤ)
53	3	nn0zd 12540	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℤ)
54	2	nnge1d 12216	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ≤ 𝐾)
55	2	nnred 12180	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ ℝ)
56	55	leidd 11707	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ≤ 𝐾)
57	52, 53, 53, 54, 56	elfzd 13460	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ (1...𝐾))
58	57	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝐾 ∈ (1...𝐾))
59		ovexd 7391	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐾 + Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙)) ∈ V)
60	47, 51, 58, 59	fvmptd 6943	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝐹‘𝑐)‘𝐾) = (𝐾 + Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙)))
61	60	oveq2d 7372	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)) = ((𝑁 + 𝐾) − (𝐾 + Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙))))
62	1	nn0cnd 12491	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℂ)
63	62	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑁 ∈ ℂ)
64	55	recnd 11164	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℂ)
65	64	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝐾 ∈ ℂ)
66	63, 65	addcomd 11339	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑁 + 𝐾) = (𝐾 + 𝑁))
67	66	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝑁 + 𝐾) − (𝐾 + Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙))) = ((𝐾 + 𝑁) − (𝐾 + Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙))))
68		1zzd 12549	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → 1 ∈ ℤ)
69	53	ad2antrr 732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → 𝐾 ∈ ℤ)
70	69	peano2zd 12627	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → (𝐾 + 1) ∈ ℤ)
71		elfzelz 13469	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ (1...𝐾) → 𝑙 ∈ ℤ)
72	71	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → 𝑙 ∈ ℤ)
73		elfzle1 13472	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ (1...𝐾) → 1 ≤ 𝑙)
74	73	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → 1 ≤ 𝑙)
75	72	zred 12624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → 𝑙 ∈ ℝ)
76	55	ad2antrr 732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → 𝐾 ∈ ℝ)
77	70	zred 12624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → (𝐾 + 1) ∈ ℝ)
78		elfzle2 13473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 ∈ (1...𝐾) → 𝑙 ≤ 𝐾)
79	78	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → 𝑙 ≤ 𝐾)
80	76	lep1d 12078	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → 𝐾 ≤ (𝐾 + 1))
81	75, 76, 77, 79, 80	letrd 11294	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → 𝑙 ≤ (𝐾 + 1))
82	68, 70, 72, 74, 81	elfzd 13460	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → 𝑙 ∈ (1...(𝐾 + 1)))
83	5	eleq2i 2831	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 ∈ 𝐴 ↔ 𝑐 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)})
84	83	bilani 505	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)})
85		vex 3435	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑐 ∈ V
86		feq1 6633	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔 = 𝑐 → (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ↔ 𝑐:(1...(𝐾 + 1))⟶ℕ₀))
87		simpl 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑔 = 𝑐 ∧ 𝑖 ∈ (1...(𝐾 + 1))) → 𝑔 = 𝑐)
88	87	fveq1d 6829	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑔 = 𝑐 ∧ 𝑖 ∈ (1...(𝐾 + 1))) → (𝑔‘𝑖) = (𝑐‘𝑖))
89	88	sumeq2dv 15655	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑔 = 𝑐 → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = Σ𝑖 ∈ (1...(𝐾 + 1))(𝑐‘𝑖))
90	89	eqeq1d 2741	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑔 = 𝑐 → (Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑐‘𝑖) = 𝑁))
91	86, 90	anbi12d 638	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑔 = 𝑐 → ((𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁) ↔ (𝑐:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑐‘𝑖) = 𝑁)))
92	85, 91	elab 3617	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ↔ (𝑐:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑐‘𝑖) = 𝑁))
93	92	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ↔ (𝑐:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑐‘𝑖) = 𝑁)))
94	93	biimpd 230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} → (𝑐:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑐‘𝑖) = 𝑁)))
95	84, 94	mpd 15	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐:(1...(𝐾 + 1))⟶ℕ₀ ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑐‘𝑖) = 𝑁))
96	95	simpld 495	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐:(1...(𝐾 + 1))⟶ℕ₀)
97	96	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → 𝑐:(1...(𝐾 + 1))⟶ℕ₀)
98	97	ffvelcdmda 7025	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) ∧ 𝑙 ∈ (1...(𝐾 + 1))) → (𝑐‘𝑙) ∈ ℕ₀)
99	82, 98	mpdan 693	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...𝐾)) → (𝑐‘𝑙) ∈ ℕ₀)
100	45, 99	fsumnn0cl 15689	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙) ∈ ℕ₀)
101	100	nn0cnd 12491	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙) ∈ ℂ)
102	65, 63, 101	pnpcand 11533	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝐾 + 𝑁) − (𝐾 + Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙))) = (𝑁 − Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙)))
103		eqid 2739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 = 1
104		1p0e1 12291	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 + 0) = 1
105	103, 104	eqtr4i 2765	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 = (1 + 0)
106	105	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 1 = (1 + 0))
107		1red 11136	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 1 ∈ ℝ)
108		0le1 11664	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ≤ 1
109	108	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 0 ≤ 1)
110	107, 11, 55, 107, 54, 109	le2addd 11760	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (1 + 0) ≤ (𝐾 + 1))
111	106, 110	eqbrtrd 5094	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 ≤ (𝐾 + 1))
112	53	peano2zd 12627	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐾 + 1) ∈ ℤ)
113		eluz 12793	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1 ∈ ℤ ∧ (𝐾 + 1) ∈ ℤ) → ((𝐾 + 1) ∈ (ℤ_≥‘1) ↔ 1 ≤ (𝐾 + 1)))
114	52, 112, 113	syl2anc 590	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐾 + 1) ∈ (ℤ_≥‘1) ↔ 1 ≤ (𝐾 + 1)))
115	111, 114	mpbird 258	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐾 + 1) ∈ (ℤ_≥‘1))
116	115	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐾 + 1) ∈ (ℤ_≥‘1))
117	96	ffvelcdmda 7025	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...(𝐾 + 1))) → (𝑐‘𝑙) ∈ ℕ₀)
118	117	nn0cnd 12491	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...(𝐾 + 1))) → (𝑐‘𝑙) ∈ ℂ)
119		fveq2 6827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = (𝐾 + 1) → (𝑐‘𝑙) = (𝑐‘(𝐾 + 1)))
120	116, 118, 119	fsumm1 15704	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → Σ𝑙 ∈ (1...(𝐾 + 1))(𝑐‘𝑙) = (Σ𝑙 ∈ (1...((𝐾 + 1) − 1))(𝑐‘𝑙) + (𝑐‘(𝐾 + 1))))
121		1cnd 11130	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 1 ∈ ℂ)
122	65, 121	pncand 11497	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝐾 + 1) − 1) = 𝐾)
123	122	oveq2d 7372	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (1...((𝐾 + 1) − 1)) = (1...𝐾))
124	123	sumeq1d 15653	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → Σ𝑙 ∈ (1...((𝐾 + 1) − 1))(𝑐‘𝑙) = Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙))
125	124	oveq1d 7371	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (Σ𝑙 ∈ (1...((𝐾 + 1) − 1))(𝑐‘𝑙) + (𝑐‘(𝐾 + 1))) = (Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙) + (𝑐‘(𝐾 + 1))))
126	120, 125	eqtrd 2774	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → Σ𝑙 ∈ (1...(𝐾 + 1))(𝑐‘𝑙) = (Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙) + (𝑐‘(𝐾 + 1))))
127	126	eqcomd 2745	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙) + (𝑐‘(𝐾 + 1))) = Σ𝑙 ∈ (1...(𝐾 + 1))(𝑐‘𝑙))
128		fveq2 6827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑖 → (𝑐‘𝑙) = (𝑐‘𝑖))
129		nfcv 2901	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖(𝑐‘𝑙)
130		nfcv 2901	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑙(𝑐‘𝑖)
131	128, 129, 130	cbvsum 15648	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑙 ∈ (1...(𝐾 + 1))(𝑐‘𝑙) = Σ𝑖 ∈ (1...(𝐾 + 1))(𝑐‘𝑖)
132	131	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → Σ𝑙 ∈ (1...(𝐾 + 1))(𝑐‘𝑙) = Σ𝑖 ∈ (1...(𝐾 + 1))(𝑐‘𝑖))
133	95	simprd 496	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑐‘𝑖) = 𝑁)
134	132, 133	eqtrd 2774	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → Σ𝑙 ∈ (1...(𝐾 + 1))(𝑐‘𝑙) = 𝑁)
135	127, 134	eqtrd 2774	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙) + (𝑐‘(𝐾 + 1))) = 𝑁)
136		1zzd 12549	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 1 ∈ ℤ)
137	53	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝐾 ∈ ℤ)
138	137	peano2zd 12627	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐾 + 1) ∈ ℤ)
139		1e0p1 12677	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 = (0 + 1)
140	139	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 1 = (0 + 1))
141		0red 11138	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 0 ∈ ℝ)
142	55	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝐾 ∈ ℝ)
143		1red 11136	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 1 ∈ ℝ)
144	11, 55, 12	ltled 11285	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 0 ≤ 𝐾)
145	144	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 0 ≤ 𝐾)
146	141, 142, 143, 145	leadd1dd 11755	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (0 + 1) ≤ (𝐾 + 1))
147	140, 146	eqbrtrd 5094	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 1 ≤ (𝐾 + 1))
148	55, 55, 107, 56	leadd1dd 11755	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐾 + 1) ≤ (𝐾 + 1))
149	148	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐾 + 1) ≤ (𝐾 + 1))
150	136, 138, 138, 147, 149	elfzd 13460	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐾 + 1) ∈ (1...(𝐾 + 1)))
151	96, 150	ffvelcdmd 7026	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘(𝐾 + 1)) ∈ ℕ₀)
152	151	nn0cnd 12491	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘(𝐾 + 1)) ∈ ℂ)
153	63, 101, 152	subaddd 11514	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝑁 − Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙)) = (𝑐‘(𝐾 + 1)) ↔ (Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙) + (𝑐‘(𝐾 + 1))) = 𝑁))
154	135, 153	mpbird 258	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑁 − Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙)) = (𝑐‘(𝐾 + 1)))
155	102, 154	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝐾 + 𝑁) − (𝐾 + Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙))) = (𝑐‘(𝐾 + 1)))
156	67, 155	eqtrd 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝑁 + 𝐾) − (𝐾 + Σ𝑙 ∈ (1...𝐾)(𝑐‘𝑙))) = (𝑐‘(𝐾 + 1)))
157	61, 156	eqtrd 2774	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)) = (𝑐‘(𝐾 + 1)))
158	157	3adant3 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)) = (𝑐‘(𝐾 + 1)))
159	158	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ 𝑘 = (𝐾 + 1)) → ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)) = (𝑐‘(𝐾 + 1)))
160		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ 𝑘 = (𝐾 + 1)) → 𝑘 = (𝐾 + 1))
161	160	fveq2d 6831	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ 𝑘 = (𝐾 + 1)) → (𝑐‘𝑘) = (𝑐‘(𝐾 + 1)))
162	161	eqcomd 2745	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ 𝑘 = (𝐾 + 1)) → (𝑐‘(𝐾 + 1)) = (𝑐‘𝑘))
163	159, 162	eqtrd 2774	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ 𝑘 = (𝐾 + 1)) → ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)) = (𝑐‘𝑘))
164	47	fveq1d 6829	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝐹‘𝑐)‘1) = ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘1))
165	164	oveq1d 7371	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (((𝐹‘𝑐)‘1) − 1) = (((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘1) − 1))
166		eqidd 2740	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙))) = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙))))
167		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑗 = 1) → 𝑗 = 1)
168	167	oveq2d 7372	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑗 = 1) → (1...𝑗) = (1...1))
169	168	sumeq1d 15653	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑗 = 1) → Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙) = Σ𝑙 ∈ (1...1)(𝑐‘𝑙))
170	167, 169	oveq12d 7374	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑗 = 1) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)) = (1 + Σ𝑙 ∈ (1...1)(𝑐‘𝑙)))
171	143	leidd 11707	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 1 ≤ 1)
172	54	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 1 ≤ 𝐾)
173	136, 137, 136, 171, 172	elfzd 13460	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 1 ∈ (1...𝐾))
174		ovexd 7391	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (1 + Σ𝑙 ∈ (1...1)(𝑐‘𝑙)) ∈ V)
175	166, 170, 173, 174	fvmptd 6943	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘1) = (1 + Σ𝑙 ∈ (1...1)(𝑐‘𝑙)))
176	175	oveq1d 7371	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘1) − 1) = ((1 + Σ𝑙 ∈ (1...1)(𝑐‘𝑙)) − 1))
177		fzfid 13926	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (1...1) ∈ Fin)
178		1zzd 12549	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → 1 ∈ ℤ)
179	137	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → 𝐾 ∈ ℤ)
180	179	peano2zd 12627	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → (𝐾 + 1) ∈ ℤ)
181		elfzelz 13469	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 ∈ (1...1) → 𝑙 ∈ ℤ)
182	181	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → 𝑙 ∈ ℤ)
183		elfzle1 13472	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 ∈ (1...1) → 1 ≤ 𝑙)
184	183	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → 1 ≤ 𝑙)
185	182	zred 12624	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → 𝑙 ∈ ℝ)
186		0red 11138	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → 0 ∈ ℝ)
187		1red 11136	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → 1 ∈ ℝ)
188	186, 187	readdcld 11165	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → (0 + 1) ∈ ℝ)
189	180	zred 12624	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → (𝐾 + 1) ∈ ℝ)
190		elfzle2 13473	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 ∈ (1...1) → 𝑙 ≤ 1)
191	190	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → 𝑙 ≤ 1)
192	139	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → 1 = (0 + 1))
193	191, 192	breqtrd 5098	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → 𝑙 ≤ (0 + 1))
194	146	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → (0 + 1) ≤ (𝐾 + 1))
195	185, 188, 189, 193, 194	letrd 11294	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → 𝑙 ≤ (𝐾 + 1))
196	178, 180, 182, 184, 195	elfzd 13460	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → 𝑙 ∈ (1...(𝐾 + 1)))
197	117	adantlr 721	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) ∧ 𝑙 ∈ (1...(𝐾 + 1))) → (𝑐‘𝑙) ∈ ℕ₀)
198	196, 197	mpdan 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑙 ∈ (1...1)) → (𝑐‘𝑙) ∈ ℕ₀)
199	177, 198	fsumnn0cl 15689	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → Σ𝑙 ∈ (1...1)(𝑐‘𝑙) ∈ ℕ₀)
200	199	nn0cnd 12491	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → Σ𝑙 ∈ (1...1)(𝑐‘𝑙) ∈ ℂ)
201	121, 200	pncan2d 11498	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((1 + Σ𝑙 ∈ (1...1)(𝑐‘𝑙)) − 1) = Σ𝑙 ∈ (1...1)(𝑐‘𝑙))
202	136, 138, 136, 171, 147	elfzd 13460	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 1 ∈ (1...(𝐾 + 1)))
203	96, 202	ffvelcdmd 7026	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘1) ∈ ℕ₀)
204	203	nn0cnd 12491	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑐‘1) ∈ ℂ)
205		fveq2 6827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 1 → (𝑐‘𝑙) = (𝑐‘1))
206	205	fsum1 15700	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℤ ∧ (𝑐‘1) ∈ ℂ) → Σ𝑙 ∈ (1...1)(𝑐‘𝑙) = (𝑐‘1))
207	136, 204, 206	syl2anc 590	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → Σ𝑙 ∈ (1...1)(𝑐‘𝑙) = (𝑐‘1))
208	201, 207	eqtrd 2774	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → ((1 + Σ𝑙 ∈ (1...1)(𝑐‘𝑙)) − 1) = (𝑐‘1))
209	176, 208	eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘1) − 1) = (𝑐‘1))
210	165, 209	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (((𝐹‘𝑐)‘1) − 1) = (𝑐‘1))
211	210	3adant3 1138	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → (((𝐹‘𝑐)‘1) − 1) = (𝑐‘1))
212	211	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → (((𝐹‘𝑐)‘1) − 1) = (𝑐‘1))
213	212	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ 𝑘 = 1) → (((𝐹‘𝑐)‘1) − 1) = (𝑐‘1))
214		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ 𝑘 = 1) → 𝑘 = 1)
215	214	fveq2d 6831	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ 𝑘 = 1) → (𝑐‘𝑘) = (𝑐‘1))
216	215	eqcomd 2745	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ 𝑘 = 1) → (𝑐‘1) = (𝑐‘𝑘))
217	213, 216	eqtrd 2774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ 𝑘 = 1) → (((𝐹‘𝑐)‘1) − 1) = (𝑐‘𝑘))
218	4	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))))
219		simpllr 781	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑎 = 𝑐) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑙 ∈ (1...𝑗)) → 𝑎 = 𝑐)
220	219	fveq1d 6829	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑎 = 𝑐) ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑙 ∈ (1...𝑗)) → (𝑎‘𝑙) = (𝑐‘𝑙))
221	220	sumeq2dv 15655	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑎 = 𝑐) ∧ 𝑗 ∈ (1...𝐾)) → Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙) = Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙))
222	221	oveq2d 7372	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑎 = 𝑐) ∧ 𝑗 ∈ (1...𝐾)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)) = (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))
223	222	mpteq2dva 5165	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑎 = 𝑐) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙))))
224		simpll2 1220	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝑐 ∈ 𝐴)
225		fzfid 13926	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (1...𝐾) ∈ Fin)
226	225	mptexd 7168	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙))) ∈ V)
227	218, 223, 224, 226	fvmptd 6943	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝐹‘𝑐) = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙))))
228	227	fveq1d 6829	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((𝐹‘𝑐)‘𝑘) = ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘𝑘))
229	227	fveq1d 6829	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((𝐹‘𝑐)‘(𝑘 − 1)) = ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘(𝑘 − 1)))
230	228, 229	oveq12d 7374	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) = (((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘𝑘) − ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘(𝑘 − 1))))
231	230	oveq1d 7371	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1) = ((((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘𝑘) − ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘(𝑘 − 1))) − 1))
232		eqidd 2740	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙))) = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙))))
233		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑗 = 𝑘) → 𝑗 = 𝑘)
234	233	oveq2d 7372	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑗 = 𝑘) → (1...𝑗) = (1...𝑘))
235	234	sumeq1d 15653	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑗 = 𝑘) → Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙) = Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙))
236	233, 235	oveq12d 7374	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑗 = 𝑘) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)) = (𝑘 + Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙)))
237		1zzd 12549	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 1 ∈ ℤ)
238	137	3adant3 1138	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → 𝐾 ∈ ℤ)
239	238	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → 𝐾 ∈ ℤ)
240	239	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝐾 ∈ ℤ)
241		elfznn 13498	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...(𝐾 + 1)) → 𝑘 ∈ ℕ)
242	241	3ad2ant3 1141	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → 𝑘 ∈ ℕ)
243	242	nnzd 12541	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → 𝑘 ∈ ℤ)
244	243	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → 𝑘 ∈ ℤ)
245	244	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ ℤ)
246	242	nnge1d 12216	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → 1 ≤ 𝑘)
247	246	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → 1 ≤ 𝑘)
248	247	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 1 ≤ 𝑘)
249		simpl3 1200	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → 𝑘 ∈ (1...(𝐾 + 1)))
250		1zzd 12549	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → 1 ∈ ℤ)
251	239	peano2zd 12627	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → (𝐾 + 1) ∈ ℤ)
252		elfz 13458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ ℤ ∧ 1 ∈ ℤ ∧ (𝐾 + 1) ∈ ℤ) → (𝑘 ∈ (1...(𝐾 + 1)) ↔ (1 ≤ 𝑘 ∧ 𝑘 ≤ (𝐾 + 1))))
253	244, 250, 251, 252	syl3anc 1379	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → (𝑘 ∈ (1...(𝐾 + 1)) ↔ (1 ≤ 𝑘 ∧ 𝑘 ≤ (𝐾 + 1))))
254	253	biimpd 230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → (𝑘 ∈ (1...(𝐾 + 1)) → (1 ≤ 𝑘 ∧ 𝑘 ≤ (𝐾 + 1))))
255	249, 254	mpd 15	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → (1 ≤ 𝑘 ∧ 𝑘 ≤ (𝐾 + 1)))
256	255	simprd 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → 𝑘 ≤ (𝐾 + 1))
257		neqne 2942	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑘 = (𝐾 + 1) → 𝑘 ≠ (𝐾 + 1))
258	257	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → 𝑘 ≠ (𝐾 + 1))
259	258	necomd 2989	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → (𝐾 + 1) ≠ 𝑘)
260	256, 259	jca 516	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → (𝑘 ≤ (𝐾 + 1) ∧ (𝐾 + 1) ≠ 𝑘))
261	244	zred 12624	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → 𝑘 ∈ ℝ)
262	251	zred 12624	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → (𝐾 + 1) ∈ ℝ)
263	261, 262	ltlend 11282	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → (𝑘 < (𝐾 + 1) ↔ (𝑘 ≤ (𝐾 + 1) ∧ (𝐾 + 1) ≠ 𝑘)))
264	260, 263	mpbird 258	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → 𝑘 < (𝐾 + 1))
265		zleltp1 12569	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑘 ≤ 𝐾 ↔ 𝑘 < (𝐾 + 1)))
266	244, 239, 265	syl2anc 590	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → (𝑘 ≤ 𝐾 ↔ 𝑘 < (𝐾 + 1)))
267	264, 266	mpbird 258	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → 𝑘 ≤ 𝐾)
268	267	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝑘 ≤ 𝐾)
269	237, 240, 245, 248, 268	elfzd 13460	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ (1...𝐾))
270		ovexd 7391	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝑘 + Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙)) ∈ V)
271	232, 236, 269, 270	fvmptd 6943	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘𝑘) = (𝑘 + Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙)))
272		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑗 = (𝑘 − 1)) → 𝑗 = (𝑘 − 1))
273	272	oveq2d 7372	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑗 = (𝑘 − 1)) → (1...𝑗) = (1...(𝑘 − 1)))
274	273	sumeq1d 15653	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑗 = (𝑘 − 1)) → Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙) = Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))
275	272, 274	oveq12d 7374	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑗 = (𝑘 − 1)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)) = ((𝑘 − 1) + Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙)))
276		1zzd 12549	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 1 ∈ ℤ)
277	53	ad2antrr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = 1) → 𝐾 ∈ ℤ)
278	277	3impa 1115	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝐾 ∈ ℤ)
279	241	nnzd 12541	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (1...(𝐾 + 1)) → 𝑘 ∈ ℤ)
280	279	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → 𝑘 ∈ ℤ)
281	280	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ ℤ)
282	281	3impa 1115	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ ℤ)
283	282, 276	zsubcld 12629	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ∈ ℤ)
284		neqne 2942	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑘 = 1 → 𝑘 ≠ 1)
285	284	3ad2ant3 1141	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝑘 ≠ 1)
286	107	3ad2ant1 1139	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 1 ∈ ℝ)
287	282	zred 12624	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ ℝ)
288		simp2 1143	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ (1...(𝐾 + 1)))
289		elfzle1 13472	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (1...(𝐾 + 1)) → 1 ≤ 𝑘)
290	288, 289	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 1 ≤ 𝑘)
291	286, 287, 290	leltned 11290	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (1 < 𝑘 ↔ 𝑘 ≠ 1))
292	285, 291	mpbird 258	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 1 < 𝑘)
293	276, 282	zltp1led 12573	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (1 < 𝑘 ↔ (1 + 1) ≤ 𝑘))
294	292, 293	mpbid 233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (1 + 1) ≤ 𝑘)
295		leaddsub 11617	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 + 1) ≤ 𝑘 ↔ 1 ≤ (𝑘 − 1)))
296	286, 286, 287, 295	syl3anc 1379	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((1 + 1) ≤ 𝑘 ↔ 1 ≤ (𝑘 − 1)))
297	294, 296	mpbid 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 1 ≤ (𝑘 − 1))
298	283	zred 12624	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ∈ ℝ)
299	55	3ad2ant1 1139	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝐾 ∈ ℝ)
300		1red 11136	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 1 ∈ ℝ)
301	299, 300	readdcld 11165	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝐾 + 1) ∈ ℝ)
302	301, 300	resubcld 11569	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((𝐾 + 1) − 1) ∈ ℝ)
303		elfzle2 13473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (1...(𝐾 + 1)) → 𝑘 ≤ (𝐾 + 1))
304	303	3ad2ant2 1140	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝑘 ≤ (𝐾 + 1))
305	287, 301, 300, 304	lesub1dd 11757	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ≤ ((𝐾 + 1) − 1))
306	64	3ad2ant1 1139	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝐾 ∈ ℂ)
307		1cnd 11130	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 1 ∈ ℂ)
308	306, 307	pncand 11497	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((𝐾 + 1) − 1) = 𝐾)
309	56	3ad2ant1 1139	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝐾 ≤ 𝐾)
310	308, 309	eqbrtrd 5094	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((𝐾 + 1) − 1) ≤ 𝐾)
311	298, 302, 299, 305, 310	letrd 11294	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ≤ 𝐾)
312	276, 278, 283, 297, 311	elfzd 13460	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ∈ (1...𝐾))
313	312	3expa 1124	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ∈ (1...𝐾))
314	313	3adantl2 1174	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ∈ (1...𝐾))
315	314	adantlr 721	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ∈ (1...𝐾))
316		ovexd 7391	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((𝑘 − 1) + Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙)) ∈ V)
317	232, 275, 315, 316	fvmptd 6943	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘(𝑘 − 1)) = ((𝑘 − 1) + Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙)))
318	271, 317	oveq12d 7374	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘𝑘) − ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘(𝑘 − 1))) = ((𝑘 + Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙)) − ((𝑘 − 1) + Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))))
319	318	oveq1d 7371	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘𝑘) − ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘(𝑘 − 1))) − 1) = (((𝑘 + Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙)) − ((𝑘 − 1) + Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))) − 1))
320	245	zcnd 12625	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ ℂ)
321		fzfid 13926	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (1...𝑘) ∈ Fin)
322		1zzd 12549	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → 1 ∈ ℤ)
323	240	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → 𝐾 ∈ ℤ)
324	323	peano2zd 12627	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → (𝐾 + 1) ∈ ℤ)
325		elfznn 13498	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ (1...𝑘) → 𝑙 ∈ ℕ)
326	325	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → 𝑙 ∈ ℕ)
327	326	nnzd 12541	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → 𝑙 ∈ ℤ)
328	326	nnge1d 12216	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → 1 ≤ 𝑙)
329	326	nnred 12180	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → 𝑙 ∈ ℝ)
330	261	ad2antrr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → 𝑘 ∈ ℝ)
331	262	ad2antrr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → (𝐾 + 1) ∈ ℝ)
332		elfzle2 13473	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ (1...𝑘) → 𝑙 ≤ 𝑘)
333	332	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → 𝑙 ≤ 𝑘)
334	256	ad2antrr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → 𝑘 ≤ (𝐾 + 1))
335	329, 330, 331, 333, 334	letrd 11294	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → 𝑙 ≤ (𝐾 + 1))
336	322, 324, 327, 328, 335	elfzd 13460	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → 𝑙 ∈ (1...(𝐾 + 1)))
337	96	3adant3 1138	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → 𝑐:(1...(𝐾 + 1))⟶ℕ₀)
338	337	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → 𝑐:(1...(𝐾 + 1))⟶ℕ₀)
339	338	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝑐:(1...(𝐾 + 1))⟶ℕ₀)
340	339	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → 𝑐:(1...(𝐾 + 1))⟶ℕ₀)
341	340	ffvelcdmda 7025	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) ∧ 𝑙 ∈ (1...(𝐾 + 1))) → (𝑐‘𝑙) ∈ ℕ₀)
342	336, 341	mpdan 693	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → (𝑐‘𝑙) ∈ ℕ₀)
343	321, 342	fsumnn0cl 15689	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) ∈ ℕ₀)
344	343	nn0cnd 12491	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) ∈ ℂ)
345		1cnd 11130	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 1 ∈ ℂ)
346	320, 345	subcld 11496	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ∈ ℂ)
347		fzfid 13926	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (1...(𝑘 − 1)) ∈ Fin)
348		1zzd 12549	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 1 ∈ ℤ)
349	240	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 𝐾 ∈ ℤ)
350	349	peano2zd 12627	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → (𝐾 + 1) ∈ ℤ)
351		elfznn 13498	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ (1...(𝑘 − 1)) → 𝑙 ∈ ℕ)
352	351	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 𝑙 ∈ ℕ)
353	352	nnzd 12541	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 𝑙 ∈ ℤ)
354	352	nnge1d 12216	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 1 ≤ 𝑙)
355	352	nnred 12180	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 𝑙 ∈ ℝ)
356	261	ad2antrr 732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 𝑘 ∈ ℝ)
357		1red 11136	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 1 ∈ ℝ)
358	356, 357	resubcld 11569	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → (𝑘 − 1) ∈ ℝ)
359	262	ad2antrr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → (𝐾 + 1) ∈ ℝ)
360		elfzle2 13473	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ (1...(𝑘 − 1)) → 𝑙 ≤ (𝑘 − 1))
361	360	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 𝑙 ≤ (𝑘 − 1))
362	356	lem1d 12080	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → (𝑘 − 1) ≤ 𝑘)
363	256	ad2antrr 732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 𝑘 ≤ (𝐾 + 1))
364	358, 356, 359, 362, 363	letrd 11294	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → (𝑘 − 1) ≤ (𝐾 + 1))
365	355, 358, 359, 361, 364	letrd 11294	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 𝑙 ≤ (𝐾 + 1))
366	348, 350, 353, 354, 365	elfzd 13460	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 𝑙 ∈ (1...(𝐾 + 1)))
367	339	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → 𝑐:(1...(𝐾 + 1))⟶ℕ₀)
368	367	ffvelcdmda 7025	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) ∧ 𝑙 ∈ (1...(𝐾 + 1))) → (𝑐‘𝑙) ∈ ℕ₀)
369	366, 368	mpdan 693	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...(𝑘 − 1))) → (𝑐‘𝑙) ∈ ℕ₀)
370	347, 369	fsumnn0cl 15689	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙) ∈ ℕ₀)
371	370	nn0cnd 12491	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙) ∈ ℂ)
372	320, 344, 346, 371	addsub4d 11543	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((𝑘 + Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙)) − ((𝑘 − 1) + Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))) = ((𝑘 − (𝑘 − 1)) + (Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))))
373	372	oveq1d 7371	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (((𝑘 + Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙)) − ((𝑘 − 1) + Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))) − 1) = (((𝑘 − (𝑘 − 1)) + (Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))) − 1))
374	320, 345	nncand 11501	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝑘 − (𝑘 − 1)) = 1)
375	374	oveq1d 7371	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((𝑘 − (𝑘 − 1)) + (Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))) = (1 + (Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))))
376	375	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (((𝑘 − (𝑘 − 1)) + (Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))) − 1) = ((1 + (Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))) − 1))
377	344, 371	subcld 11496	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙)) ∈ ℂ)
378	345, 377	pncan2d 11498	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((1 + (Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))) − 1) = (Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙)))
379	136	3adant3 1138	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → 1 ∈ ℤ)
380	379, 243, 246	3jca 1134	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → (1 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 1 ≤ 𝑘))
381		eluz2 12785	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 1 ≤ 𝑘))
382	380, 381	sylibr 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → 𝑘 ∈ (ℤ_≥‘1))
383	382	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → 𝑘 ∈ (ℤ_≥‘1))
384	383	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ (ℤ_≥‘1))
385	342	nn0cnd 12491	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) ∧ 𝑙 ∈ (1...𝑘)) → (𝑐‘𝑙) ∈ ℂ)
386		fveq2 6827	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑘 → (𝑐‘𝑙) = (𝑐‘𝑘))
387	384, 385, 386	fsumm1 15704	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) = (Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙) + (𝑐‘𝑘)))
388	387	eqcomd 2745	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙) + (𝑐‘𝑘)) = Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙))
389		simp3 1144	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → 𝑘 ∈ (1...(𝐾 + 1)))
390	337, 389	ffvelcdmd 7026	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → (𝑐‘𝑘) ∈ ℕ₀)
391	390	nn0cnd 12491	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → (𝑐‘𝑘) ∈ ℂ)
392	391	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → (𝑐‘𝑘) ∈ ℂ)
393	392	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (𝑐‘𝑘) ∈ ℂ)
394	344, 371, 393	subaddd 11514	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙)) = (𝑐‘𝑘) ↔ (Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙) + (𝑐‘𝑘)) = Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙)))
395	388, 394	mpbird 258	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙)) = (𝑐‘𝑘))
396	378, 395	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((1 + (Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))) − 1) = (𝑐‘𝑘))
397	376, 396	eqtrd 2774	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (((𝑘 − (𝑘 − 1)) + (Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙) − Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))) − 1) = (𝑐‘𝑘))
398	373, 397	eqtrd 2774	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → (((𝑘 + Σ𝑙 ∈ (1...𝑘)(𝑐‘𝑙)) − ((𝑘 − 1) + Σ𝑙 ∈ (1...(𝑘 − 1))(𝑐‘𝑙))) − 1) = (𝑐‘𝑘))
399	319, 398	eqtrd 2774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘𝑘) − ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑐‘𝑙)))‘(𝑘 − 1))) − 1) = (𝑐‘𝑘))
400	231, 399	eqtrd 2774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) ∧ ¬ 𝑘 = 1) → ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1) = (𝑐‘𝑘))
401	217, 400	ifeqda 4491	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) ∧ ¬ 𝑘 = (𝐾 + 1)) → if(𝑘 = 1, (((𝐹‘𝑐)‘1) − 1), ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1)) = (𝑐‘𝑘))
402	163, 401	ifeqda 4491	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴 ∧ 𝑘 ∈ (1...(𝐾 + 1))) → if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)), if(𝑘 = 1, (((𝐹‘𝑐)‘1) − 1), ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1))) = (𝑐‘𝑘))
403	402	3expa 1124	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐴) ∧ 𝑘 ∈ (1...(𝐾 + 1))) → if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)), if(𝑘 = 1, (((𝐹‘𝑐)‘1) − 1), ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1))) = (𝑐‘𝑘))
404	403	mpteq2dva 5165	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)), if(𝑘 = 1, (((𝐹‘𝑐)‘1) − 1), ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1)))) = (𝑘 ∈ (1...(𝐾 + 1)) ↦ (𝑐‘𝑘)))
405	96	ffnd 6656	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 Fn (1...(𝐾 + 1)))
406		dffn5 6885	. . . . . . . 8 ⊢ (𝑐 Fn (1...(𝐾 + 1)) ↔ 𝑐 = (𝑘 ∈ (1...(𝐾 + 1)) ↦ (𝑐‘𝑘)))
407	406	biimpi 217	. . . . . . 7 ⊢ (𝑐 Fn (1...(𝐾 + 1)) → 𝑐 = (𝑘 ∈ (1...(𝐾 + 1)) ↦ (𝑐‘𝑘)))
408	405, 407	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → 𝑐 = (𝑘 ∈ (1...(𝐾 + 1)) ↦ (𝑐‘𝑘)))
409	408	eqcomd 2745	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑘 ∈ (1...(𝐾 + 1)) ↦ (𝑐‘𝑘)) = 𝑐)
410	404, 409	eqtrd 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − ((𝐹‘𝑐)‘𝐾)), if(𝑘 = 1, (((𝐹‘𝑐)‘1) − 1), ((((𝐹‘𝑐)‘𝑘) − ((𝐹‘𝑐)‘(𝑘 − 1))) − 1)))) = 𝑐)
411	37, 410	eqtrd 2774	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐴) → (𝐺‘(𝐹‘𝑐)) = 𝑐)
412	411	ralrimiva 3131	. 2 ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝐺‘(𝐹‘𝑐)) = 𝑐)
413	1, 2, 4, 8, 5, 6	sticksstones12a 42642	. 2 ⊢ (𝜑 → ∀𝑑 ∈ 𝐵 (𝐹‘(𝐺‘𝑑)) = 𝑑)
414	7, 9, 412, 413	2fvidf1od 7242	1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 {cab 2717 ≠ wne 2934 ∀wral 3053 Vcvv 3431 ifcif 4454 {csn 4555 ⟨cop 4561 class class class wbr 5072 ↦ cmpt 5153 Fn wfn 6480 ⟶wf 6481 –1-1-onto→wf1o 6484 ‘cfv 6485 (class class class)co 7356 Fincfn 8883 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 < clt 11170 ≤ cle 11171 − cmin 11368 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 ℤ_≥cuz 12779 ...cfz 13452 Σcsu 15639

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-ico 13295 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-sum 15640

This theorem is referenced by: sticksstones13 42644

W3C validator