Theorem caonncan 7741

Description: Transfer nncan 11538-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	ffvelcdmda 7104	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐴‘𝑧) ∈ 𝑆)
3		caonncan.b	. . . . 5 ⊢ (𝜑 → 𝐵:𝐼⟶𝑆)
4	3	ffvelcdmda 7104	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐵‘𝑧) ∈ 𝑆)
5		caonncan.z	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
6	5	ralrimivva 3202	. . . . 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 7438	. . . . . . 7 ⊢ (𝑥 = (𝐴‘𝑧) → (𝑥𝑀𝑦) = ((𝐴‘𝑧)𝑀𝑦))
10	8, 9	oveq12d 7449	. . . . . 6 ⊢ (𝑥 = (𝐴‘𝑧) → (𝑥𝑀(𝑥𝑀𝑦)) = ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀𝑦)))
11	10	eqeq1d 2739	. . . . 5 ⊢ (𝑥 = (𝐴‘𝑧) → ((𝑥𝑀(𝑥𝑀𝑦)) = 𝑦 ↔ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀𝑦)) = 𝑦))
12		oveq2 7439	. . . . . . 7 ⊢ (𝑦 = (𝐵‘𝑧) → ((𝐴‘𝑧)𝑀𝑦) = ((𝐴‘𝑧)𝑀(𝐵‘𝑧)))
13	12	oveq2d 7447	. . . . . 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 2753	. . . . 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 3634	. . . 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 838	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧))) = (𝐵‘𝑧))
18	17	mpteq2dva 5242	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧)))) = (𝑧 ∈ 𝐼 ↦ (𝐵‘𝑧)))
19		caonncan.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
20		fvexd 6921	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐴‘𝑧) ∈ V)
21		ovexd 7466	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ((𝐴‘𝑧)𝑀(𝐵‘𝑧)) ∈ V)
22	1	feqmptd 6977	. . 3 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝐼 ↦ (𝐴‘𝑧)))
23		fvexd 6921	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐵‘𝑧) ∈ V)
24	3	feqmptd 6977	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑧 ∈ 𝐼 ↦ (𝐵‘𝑧)))
25	19, 20, 23, 22, 24	offval2 7717	. . 3 ⊢ (𝜑 → (𝐴 ∘f 𝑀𝐵) = (𝑧 ∈ 𝐼 ↦ ((𝐴‘𝑧)𝑀(𝐵‘𝑧))))
26	19, 20, 21, 22, 25	offval2 7717	. 2 ⊢ (𝜑 → (𝐴 ∘f 𝑀(𝐴 ∘f 𝑀𝐵)) = (𝑧 ∈ 𝐼 ↦ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧)))))
27	18, 26, 24	3eqtr4d 2787	1 ⊢ (𝜑 → (𝐴 ∘f 𝑀(𝐴 ∘f 𝑀𝐵)) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480   ↦ cmpt 5225  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∘_f cof 7695

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697

This theorem is referenced by:  psropprmul  22239