Theorem caonncan 7649

Description: Transfer nncan 11385-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 7012	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐴‘𝑧) ∈ 𝑆)
3		caonncan.b	. . . . 5 ⊢ (𝜑 → 𝐵:𝐼⟶𝑆)
4	3	ffvelcdmda 7012	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐵‘𝑧) ∈ 𝑆)
5		caonncan.z	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
6	5	ralrimivva 3175	. . . . 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 7348	. . . . . . 7 ⊢ (𝑥 = (𝐴‘𝑧) → (𝑥𝑀𝑦) = ((𝐴‘𝑧)𝑀𝑦))
10	8, 9	oveq12d 7359	. . . . . 6 ⊢ (𝑥 = (𝐴‘𝑧) → (𝑥𝑀(𝑥𝑀𝑦)) = ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀𝑦)))
11	10	eqeq1d 2733	. . . . 5 ⊢ (𝑥 = (𝐴‘𝑧) → ((𝑥𝑀(𝑥𝑀𝑦)) = 𝑦 ↔ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀𝑦)) = 𝑦))
12		oveq2 7349	. . . . . . 7 ⊢ (𝑦 = (𝐵‘𝑧) → ((𝐴‘𝑧)𝑀𝑦) = ((𝐴‘𝑧)𝑀(𝐵‘𝑧)))
13	12	oveq2d 7357	. . . . . 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 2747	. . . . 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 3584	. . . 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 837	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧))) = (𝐵‘𝑧))
18	17	mpteq2dva 5179	. 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 6832	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐴‘𝑧) ∈ V)
21		ovexd 7376	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ((𝐴‘𝑧)𝑀(𝐵‘𝑧)) ∈ V)
22	1	feqmptd 6885	. . 3 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝐼 ↦ (𝐴‘𝑧)))
23		fvexd 6832	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐵‘𝑧) ∈ V)
24	3	feqmptd 6885	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑧 ∈ 𝐼 ↦ (𝐵‘𝑧)))
25	19, 20, 23, 22, 24	offval2 7625	. . 3 ⊢ (𝜑 → (𝐴 ∘f 𝑀𝐵) = (𝑧 ∈ 𝐼 ↦ ((𝐴‘𝑧)𝑀(𝐵‘𝑧))))
26	19, 20, 21, 22, 25	offval2 7625	. 2 ⊢ (𝜑 → (𝐴 ∘f 𝑀(𝐴 ∘f 𝑀𝐵)) = (𝑧 ∈ 𝐼 ↦ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧)))))
27	18, 26, 24	3eqtr4d 2776	1 ⊢ (𝜑 → (𝐴 ∘f 𝑀(𝐴 ∘f 𝑀𝐵)) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ↦ cmpt 5167  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341   ∘_f cof 7603

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605

This theorem is referenced by:  psropprmul  22145