Theorem caonncan 6833

Description: Transfer nncan 10189-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Hypotheses

Ref	Expression
caonncan.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
caonncan.a	⊢ (𝜑 → 𝐴:𝐼⟶𝑆)
caonncan.b	⊢ (𝜑 → 𝐵:𝐼⟶𝑆)
caonncan.z	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)

Assertion

Ref	Expression
caonncan	⊢ (𝜑 → (𝐴 ∘𝑓 𝑀(𝐴 ∘𝑓 𝑀𝐵)) = 𝐵)

Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐵(𝑥)   𝐼(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem caonncan

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caonncan.a	. . . . 5 ⊢ (𝜑 → 𝐴:𝐼⟶𝑆)
2	1	ffvelrnda 6267	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐴‘𝑧) ∈ 𝑆)
3		caonncan.b	. . . . 5 ⊢ (𝜑 → 𝐵:𝐼⟶𝑆)
4	3	ffvelrnda 6267	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐵‘𝑧) ∈ 𝑆)
5		caonncan.z	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
6	5	ralrimivva 2954	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
8		id 22	. . . . . . 7 ⊢ (𝑥 = (𝐴‘𝑧) → 𝑥 = (𝐴‘𝑧))
9		oveq1 6556	. . . . . . 7 ⊢ (𝑥 = (𝐴‘𝑧) → (𝑥𝑀𝑦) = ((𝐴‘𝑧)𝑀𝑦))
10	8, 9	oveq12d 6567	. . . . . 6 ⊢ (𝑥 = (𝐴‘𝑧) → (𝑥𝑀(𝑥𝑀𝑦)) = ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀𝑦)))
11	10	eqeq1d 2612	. . . . 5 ⊢ (𝑥 = (𝐴‘𝑧) → ((𝑥𝑀(𝑥𝑀𝑦)) = 𝑦 ↔ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀𝑦)) = 𝑦))
12		oveq2 6557	. . . . . . 7 ⊢ (𝑦 = (𝐵‘𝑧) → ((𝐴‘𝑧)𝑀𝑦) = ((𝐴‘𝑧)𝑀(𝐵‘𝑧)))
13	12	oveq2d 6565	. . . . . 6 ⊢ (𝑦 = (𝐵‘𝑧) → ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀𝑦)) = ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧))))
14		id 22	. . . . . 6 ⊢ (𝑦 = (𝐵‘𝑧) → 𝑦 = (𝐵‘𝑧))
15	13, 14	eqeq12d 2625	. . . . 5 ⊢ (𝑦 = (𝐵‘𝑧) → (((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀𝑦)) = 𝑦 ↔ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧))) = (𝐵‘𝑧)))
16	11, 15	rspc2va 3294	. . . 4 ⊢ ((((𝐴‘𝑧) ∈ 𝑆 ∧ (𝐵‘𝑧) ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦) → ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧))) = (𝐵‘𝑧))
17	2, 4, 7, 16	syl21anc 1317	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧))) = (𝐵‘𝑧))
18	17	mpteq2dva 4672	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧)))) = (𝑧 ∈ 𝐼 ↦ (𝐵‘𝑧)))
19		caonncan.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
20		fvex 6113	. . . 4 ⊢ (𝐴‘𝑧) ∈ V
21	20	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐴‘𝑧) ∈ V)
22		ovex 6577	. . . 4 ⊢ ((𝐴‘𝑧)𝑀(𝐵‘𝑧)) ∈ V
23	22	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ((𝐴‘𝑧)𝑀(𝐵‘𝑧)) ∈ V)
24	1	feqmptd 6159	. . 3 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝐼 ↦ (𝐴‘𝑧)))
25		fvex 6113	. . . . 5 ⊢ (𝐵‘𝑧) ∈ V
26	25	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐵‘𝑧) ∈ V)
27	3	feqmptd 6159	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑧 ∈ 𝐼 ↦ (𝐵‘𝑧)))
28	19, 21, 26, 24, 27	offval2 6812	. . 3 ⊢ (𝜑 → (𝐴 ∘𝑓 𝑀𝐵) = (𝑧 ∈ 𝐼 ↦ ((𝐴‘𝑧)𝑀(𝐵‘𝑧))))
29	19, 21, 23, 24, 28	offval2 6812	. 2 ⊢ (𝜑 → (𝐴 ∘𝑓 𝑀(𝐴 ∘𝑓 𝑀𝐵)) = (𝑧 ∈ 𝐼 ↦ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧)))))
30	18, 29, 27	3eqtr4d 2654	1 ⊢ (𝜑 → (𝐴 ∘𝑓 𝑀(𝐴 ∘𝑓 𝑀𝐵)) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795

This theorem is referenced by:  psropprmul  19429