Theorem sticksstones8 42148

Description: Establish mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 1-Oct-2024.)

Hypotheses

Ref	Expression
sticksstones8.1	⊢ (𝜑 → 𝑁 ∈ ℕ0)
sticksstones8.2	⊢ (𝜑 → 𝐾 ∈ ℕ0)
sticksstones8.3	⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))
sticksstones8.4	⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}
sticksstones8.5	⊢ 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}

Assertion

Ref	Expression
sticksstones8	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Distinct variable groups:   𝐴,𝑎,𝑗,𝑙,𝑥,𝑦   𝐵,𝑎   𝑗,𝐾,𝑙,𝑓,𝑥,𝑦   𝑔,𝐾,𝑖   𝑓,𝑁,𝑗   𝑔,𝑁   𝑓,𝑎   𝑔,𝑎,𝑖   𝜑,𝑎,𝑗,𝑙   𝑖,𝑙   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑓,𝑔,𝑖)   𝐴(𝑓,𝑔,𝑖)   𝐵(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑙)   𝐹(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑎,𝑙)   𝐾(𝑎)   𝑁(𝑥,𝑦,𝑖,𝑎,𝑙)

Proof of Theorem sticksstones8

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2731	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) → (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎‘𝑙))) = (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎‘𝑙))))
2		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → 𝑒 = 𝑗)
3	2	oveq2d 7406	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → (1...𝑒) = (1...𝑗))
4	3	sumeq1d 15673	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → Σ𝑙 ∈ (1...𝑒)(𝑎‘𝑙) = Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))
5	2, 4	oveq12d 7408	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎‘𝑙)) = (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))
6		simp3 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) → 𝑗 ∈ (1...𝐾))
7		ovexd 7425	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)) ∈ V)
8	1, 5, 6, 7	fvmptd 6978	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) → ((𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎‘𝑙)))‘𝑗) = (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))
9		sticksstones8.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
10	9	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) → 𝑁 ∈ ℕ0)
11		sticksstones8.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
12	11	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) → 𝐾 ∈ ℕ0)
13		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
14		sticksstones8.4	. . . . . . . . . . . . . . 15 ⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}
15	14	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)})
16	15	eqcomd 2736	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} = 𝐴)
17	13, 16	eleqtrrd 2832	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)})
18		feq1 6669	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑎 → (𝑔:(1...(𝐾 + 1))⟶ℕ0 ↔ 𝑎:(1...(𝐾 + 1))⟶ℕ0))
19		simpl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔 = 𝑎 ∧ 𝑖 ∈ (1...(𝐾 + 1))) → 𝑔 = 𝑎)
20	19	fveq1d 6863	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔 = 𝑎 ∧ 𝑖 ∈ (1...(𝐾 + 1))) → (𝑔‘𝑖) = (𝑎‘𝑖))
21	20	sumeq2dv 15675	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑎 → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖))
22	21	eqeq1d 2732	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑎 → (Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖) = 𝑁))
23	18, 22	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑎 → ((𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁) ↔ (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖) = 𝑁)))
24	23	elabg 3646	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝐴 → (𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ↔ (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖) = 𝑁)))
25	13, 24	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ↔ (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖) = 𝑁)))
26	25	biimpd 229	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} → (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖) = 𝑁)))
27	17, 26	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖) = 𝑁))
28	27	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
29	28	3adant3 1132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
30		eqid 2730	. . . . . . . . 9 ⊢ (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎‘𝑙))) = (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎‘𝑙)))
31		fveq2 6861	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑙 → (𝑎‘𝑖) = (𝑎‘𝑙))
32		nfcv 2892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑙(𝑎‘𝑖)
33		nfcv 2892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑎‘𝑙)
34	31, 32, 33	cbvsum 15668	. . . . . . . . . . 11 ⊢ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖) = Σ𝑙 ∈ (1...(𝐾 + 1))(𝑎‘𝑙)
35	27	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖) = 𝑁)
36	34, 35	eqtr3id 2779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑙 ∈ (1...(𝐾 + 1))(𝑎‘𝑙) = 𝑁)
37	36	3adant3 1132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) → Σ𝑙 ∈ (1...(𝐾 + 1))(𝑎‘𝑙) = 𝑁)
38	10, 12, 29, 6, 30, 37	sticksstones7 42147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) → ((𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎‘𝑙)))‘𝑗) ∈ (1...(𝑁 + 𝐾)))
39	8, 38	eqeltrrd 2830	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)) ∈ (1...(𝑁 + 𝐾)))
40	39	3expa 1118	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑗 ∈ (1...𝐾)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)) ∈ (1...(𝑁 + 𝐾)))
41		eqid 2730	. . . . . 6 ⊢ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))
42	40, 41	fmptd 7089	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾)))
43	9	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) → 𝑁 ∈ ℕ0)
44	43	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑁 ∈ ℕ0)
45	11	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) → 𝐾 ∈ ℕ0)
46	45	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝐾 ∈ ℕ0)
47	24	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} ↔ (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖) = 𝑁)))
48	47	biimpd 229	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} → (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖) = 𝑁)))
49	17, 48	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖) = 𝑁))
50	49	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
51	50	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
52	51	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
53	52	adantr 480	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑎:(1...(𝐾 + 1))⟶ℕ0)
54		simpllr 775	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (1...𝐾))
55		simplr 768	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (1...𝐾))
56		simpr 484	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦)
57	44, 46, 53, 54, 55, 56, 41	sticksstones6 42146	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) ∧ 𝑥 < 𝑦) → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑦))
58	57	ex 412	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) ∧ 𝑦 ∈ (1...𝐾)) → (𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑦)))
59	58	ralrimiva 3126	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ (1...𝐾)) → ∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑦)))
60	59	ralrimiva 3126	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑦)))
61	42, 60	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑦))))
62		fzfid 13945	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ∈ Fin)
63	42, 62	fexd 7204	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ V)
64		feq1 6669	. . . . . . 7 ⊢ (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) → (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ↔ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾))))
65		fveq1 6860	. . . . . . . . . 10 ⊢ (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) → (𝑓‘𝑥) = ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑥))
66		fveq1 6860	. . . . . . . . . 10 ⊢ (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) → (𝑓‘𝑦) = ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑦))
67	65, 66	breq12d 5123	. . . . . . . . 9 ⊢ (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) → ((𝑓‘𝑥) < (𝑓‘𝑦) ↔ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑦)))
68	67	imbi2d 340	. . . . . . . 8 ⊢ (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) → ((𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ (𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑦))))
69	68	2ralbidv 3202	. . . . . . 7 ⊢ (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) → (∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)) ↔ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑦))))
70	64, 69	anbi12d 632	. . . . . 6 ⊢ (𝑓 = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) → ((𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦))) ↔ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑦)))))
71	70	elabg 3646	. . . . 5 ⊢ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ V → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ↔ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑦)))))
72	63, 71	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))} ↔ ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))):(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑥) < ((𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))‘𝑦)))))
73	61, 72	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
74		sticksstones8.5	. . . 4 ⊢ 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}
75	74	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))})
76	73, 75	eleqtrrd 2832	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ 𝐵)
77		sticksstones8.3	. 2 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))
78	76, 77	fmptd 7089	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045  Vcvv 3450   class class class wbr 5110   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Fincfn 8921  1c1 11076   + caddc 11078   < clt 11215  ℕ₀cn0 12449  ...cfz 13475  Σcsu 15659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-rp 12959  df-ico 13319  df-fz 13476  df-fzo 13623  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-sum 15660

This theorem is referenced by:  sticksstones11  42151  sticksstones12  42153