Theorem caonncan 6935

Description: Transfer nncan 10310-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 6359	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐴‘𝑧) ∈ 𝑆)
3		caonncan.b	. . . . 5 ⊢ (𝜑 → 𝐵:𝐼⟶𝑆)
4	3	ffvelrnda 6359	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐵‘𝑧) ∈ 𝑆)
5		caonncan.z	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
6	5	ralrimivva 2971	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
8		id 22	. . . . . . 7 ⊢ (𝑥 = (𝐴‘𝑧) → 𝑥 = (𝐴‘𝑧))
9		oveq1 6657	. . . . . . 7 ⊢ (𝑥 = (𝐴‘𝑧) → (𝑥𝑀𝑦) = ((𝐴‘𝑧)𝑀𝑦))
10	8, 9	oveq12d 6668	. . . . . 6 ⊢ (𝑥 = (𝐴‘𝑧) → (𝑥𝑀(𝑥𝑀𝑦)) = ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀𝑦)))
11	10	eqeq1d 2624	. . . . 5 ⊢ (𝑥 = (𝐴‘𝑧) → ((𝑥𝑀(𝑥𝑀𝑦)) = 𝑦 ↔ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀𝑦)) = 𝑦))
12		oveq2 6658	. . . . . . 7 ⊢ (𝑦 = (𝐵‘𝑧) → ((𝐴‘𝑧)𝑀𝑦) = ((𝐴‘𝑧)𝑀(𝐵‘𝑧)))
13	12	oveq2d 6666	. . . . . 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 2637	. . . . 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 3323	. . . 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 1325	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧))) = (𝐵‘𝑧))
18	17	mpteq2dva 4744	. 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 6203	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐴‘𝑧) ∈ V)
21		ovexd 6680	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → ((𝐴‘𝑧)𝑀(𝐵‘𝑧)) ∈ V)
22	1	feqmptd 6249	. . 3 ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝐼 ↦ (𝐴‘𝑧)))
23		fvexd 6203	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐼) → (𝐵‘𝑧) ∈ V)
24	3	feqmptd 6249	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑧 ∈ 𝐼 ↦ (𝐵‘𝑧)))
25	19, 20, 23, 22, 24	offval2 6914	. . 3 ⊢ (𝜑 → (𝐴 ∘𝑓 𝑀𝐵) = (𝑧 ∈ 𝐼 ↦ ((𝐴‘𝑧)𝑀(𝐵‘𝑧))))
26	19, 20, 21, 22, 25	offval2 6914	. 2 ⊢ (𝜑 → (𝐴 ∘𝑓 𝑀(𝐴 ∘𝑓 𝑀𝐵)) = (𝑧 ∈ 𝐼 ↦ ((𝐴‘𝑧)𝑀((𝐴‘𝑧)𝑀(𝐵‘𝑧)))))
27	18, 26, 24	3eqtr4d 2666	1 ⊢ (𝜑 → (𝐴 ∘𝑓 𝑀(𝐴 ∘𝑓 𝑀𝐵)) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897

This theorem is referenced by:  psropprmul  19608