Theorem sticksstones8 42166

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 2736	. . . . . . . . 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 7421	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → (1...𝑒) = (1...𝑗))
4	3	sumeq1d 15716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) ∧ 𝑒 = 𝑗) → Σ𝑙 ∈ (1...𝑒)(𝑎‘𝑙) = Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))
5	2, 4	oveq12d 7423	. . . . . . . . 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 7440	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) → (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)) ∈ V)
8	1, 5, 6, 7	fvmptd 6993	. . . . . . . 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 2741	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)} = 𝐴)
17	13, 16	eleqtrrd 2837	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)})
18		feq1 6686	. . . . . . . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔 = 𝑎 ∧ 𝑖 ∈ (1...(𝐾 + 1))) → (𝑔‘𝑖) = (𝑎‘𝑖))
21	20	sumeq2dv 15718	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑎 → Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = Σ𝑖 ∈ (1...(𝐾 + 1))(𝑎‘𝑖))
22	21	eqeq1d 2737	. . . . . . . . . . . . . . . 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 3655	. . . . . . . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎‘𝑙))) = (𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎‘𝑙)))
31		fveq2 6876	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑙 → (𝑎‘𝑖) = (𝑎‘𝑙))
32		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑙(𝑎‘𝑖)
33		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑎‘𝑙)
34	31, 32, 33	cbvsum 15711	. . . . . . . . . . 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 2784	. . . . . . . . . 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 42165	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑗 ∈ (1...𝐾)) → ((𝑒 ∈ (1...𝐾) ↦ (𝑒 + Σ𝑙 ∈ (1...𝑒)(𝑎‘𝑙)))‘𝑗) ∈ (1...(𝑁 + 𝐾)))
39	8, 38	eqeltrrd 2835	. . . . . . 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 2735	. . . . . 6 ⊢ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) = (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙)))
42	40, 41	fmptd 7104	. . . . 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 42164	. . . . . . . 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 3132	. . . . . 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 3132	. . . . 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 13991	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ∈ Fin)
63	42, 62	fexd 7219	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))) ∈ V)
64		feq1 6686	. . . . . . 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 6875	. . . . . . . . . 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 6875	. . . . . . . . . 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 5132	. . . . . . . . 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 3205	. . . . . . 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 3655	. . . . 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 2837	. 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 7104	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  {cab 2713  ∀wral 3051  Vcvv 3459   class class class wbr 5119   ↦ cmpt 5201  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  Fincfn 8959  1c1 11130   + caddc 11132   < clt 11269  ℕ₀cn0 12501  ...cfz 13524  Σcsu 15702

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9454  df-oi 9524  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-3 12304  df-n0 12502  df-z 12589  df-uz 12853  df-rp 13009  df-ico 13368  df-fz 13525  df-fzo 13672  df-seq 14020  df-exp 14080  df-hash 14349  df-cj 15118  df-re 15119  df-im 15120  df-sqrt 15254  df-abs 15255  df-clim 15504  df-sum 15703

This theorem is referenced by:  sticksstones11  42169  sticksstones12  42171