Theorem caonncan 7447

Description: Transfer nncan 10915-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 6851	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐴‘𝑧) ∈ 𝑆)
3		caonncan.b	. . . . 5 ⊢ (𝜑 → 𝐵:𝐼⟶𝑆)
4	3	ffvelrnda 6851	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐵‘𝑧) ∈ 𝑆)
5		caonncan.z	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
6	5	ralrimivva 3191	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
7	6	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
8		id 22	. . . . . . 7 ⊢ (𝑥 = (𝐴‘𝑧) → 𝑥 = (𝐴‘𝑧))
9		oveq1 7163	. . . . . . 7 ⊢ (𝑥 = (𝐴‘𝑧) → (𝑥𝑀𝑦) = ((𝐴‘𝑧)𝑀𝑦))
10	8, 9	oveq12d 7174	. . . . . 6 ⊢ (𝑥 = (𝐴‘𝑧) → (𝑥𝑀(𝑥𝑀𝑦)) = ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀𝑦)))
11	10	eqeq1d 2823	. . . . 5 ⊢ (𝑥 = (𝐴‘𝑧) → ((𝑥𝑀(𝑥𝑀𝑦)) = 𝑦 ↔ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀𝑦)) = 𝑦))
12		oveq2 7164	. . . . . . 7 ⊢ (𝑦 = (𝐵‘𝑧) → ((𝐴‘𝑧)𝑀𝑦) = ((𝐴‘𝑧)𝑀(𝐵‘𝑧)))
13	12	oveq2d 7172	. . . . . 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 2837	. . . . 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 835	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧))) = (𝐵‘𝑧))
18	17	mpteq2dva 5161	. 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 6685	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐴‘𝑧) ∈ V)
21		ovexd 7191	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ((𝐴‘𝑧)𝑀(𝐵‘𝑧)) ∈ V)
22	1	feqmptd 6733	. . 3 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝐼 ↦ (𝐴‘𝑧)))
23		fvexd 6685	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐵‘𝑧) ∈ V)
24	3	feqmptd 6733	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑧 ∈ 𝐼 ↦ (𝐵‘𝑧)))
25	19, 20, 23, 22, 24	offval2 7426	. . 3 ⊢ (𝜑 → (𝐴 ∘f 𝑀𝐵) = (𝑧 ∈ 𝐼 ↦ ((𝐴‘𝑧)𝑀(𝐵‘𝑧))))
26	19, 20, 21, 22, 25	offval2 7426	. 2 ⊢ (𝜑 → (𝐴 ∘f 𝑀(𝐴 ∘f 𝑀𝐵)) = (𝑧 ∈ 𝐼 ↦ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧)))))
27	18, 26, 24	3eqtr4d 2866	1 ⊢ (𝜑 → (𝐴 ∘f 𝑀(𝐴 ∘f 𝑀𝐵)) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ↦ cmpt 5146  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   ∘_f cof 7407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409

This theorem is referenced by:  psropprmul  20406