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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 (𝜑𝐼𝑉)
caonncan.a (𝜑𝐴:𝐼𝑆)
caonncan.b (𝜑𝐵:𝐼𝑆)
caonncan.z ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
Assertion
Ref Expression
caonncan (𝜑 → (𝐴𝑓 𝑀(𝐴𝑓 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.
StepHypRef Expression
1 caonncan.a . . . . 5 (𝜑𝐴:𝐼𝑆)
21ffvelrnda 6267 . . . 4 ((𝜑𝑧𝐼) → (𝐴𝑧) ∈ 𝑆)
3 caonncan.b . . . . 5 (𝜑𝐵:𝐼𝑆)
43ffvelrnda 6267 . . . 4 ((𝜑𝑧𝐼) → (𝐵𝑧) ∈ 𝑆)
5 caonncan.z . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
65ralrimivva 2954 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
76adantr 480 . . . 4 ((𝜑𝑧𝐼) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
8 id 22 . . . . . . 7 (𝑥 = (𝐴𝑧) → 𝑥 = (𝐴𝑧))
9 oveq1 6556 . . . . . . 7 (𝑥 = (𝐴𝑧) → (𝑥𝑀𝑦) = ((𝐴𝑧)𝑀𝑦))
108, 9oveq12d 6567 . . . . . 6 (𝑥 = (𝐴𝑧) → (𝑥𝑀(𝑥𝑀𝑦)) = ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)))
1110eqeq1d 2612 . . . . 5 (𝑥 = (𝐴𝑧) → ((𝑥𝑀(𝑥𝑀𝑦)) = 𝑦 ↔ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = 𝑦))
12 oveq2 6557 . . . . . . 7 (𝑦 = (𝐵𝑧) → ((𝐴𝑧)𝑀𝑦) = ((𝐴𝑧)𝑀(𝐵𝑧)))
1312oveq2d 6565 . . . . . 6 (𝑦 = (𝐵𝑧) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))))
14 id 22 . . . . . 6 (𝑦 = (𝐵𝑧) → 𝑦 = (𝐵𝑧))
1513, 14eqeq12d 2625 . . . . 5 (𝑦 = (𝐵𝑧) → (((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = 𝑦 ↔ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧)))
1611, 15rspc2va 3294 . . . 4 ((((𝐴𝑧) ∈ 𝑆 ∧ (𝐵𝑧) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧))
172, 4, 7, 16syl21anc 1317 . . 3 ((𝜑𝑧𝐼) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧))
1817mpteq2dva 4672 . 2 (𝜑 → (𝑧𝐼 ↦ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧)))) = (𝑧𝐼 ↦ (𝐵𝑧)))
19 caonncan.i . . 3 (𝜑𝐼𝑉)
20 fvex 6113 . . . 4 (𝐴𝑧) ∈ V
2120a1i 11 . . 3 ((𝜑𝑧𝐼) → (𝐴𝑧) ∈ V)
22 ovex 6577 . . . 4 ((𝐴𝑧)𝑀(𝐵𝑧)) ∈ V
2322a1i 11 . . 3 ((𝜑𝑧𝐼) → ((𝐴𝑧)𝑀(𝐵𝑧)) ∈ V)
241feqmptd 6159 . . 3 (𝜑𝐴 = (𝑧𝐼 ↦ (𝐴𝑧)))
25 fvex 6113 . . . . 5 (𝐵𝑧) ∈ V
2625a1i 11 . . . 4 ((𝜑𝑧𝐼) → (𝐵𝑧) ∈ V)
273feqmptd 6159 . . . 4 (𝜑𝐵 = (𝑧𝐼 ↦ (𝐵𝑧)))
2819, 21, 26, 24, 27offval2 6812 . . 3 (𝜑 → (𝐴𝑓 𝑀𝐵) = (𝑧𝐼 ↦ ((𝐴𝑧)𝑀(𝐵𝑧))))
2919, 21, 23, 24, 28offval2 6812 . 2 (𝜑 → (𝐴𝑓 𝑀(𝐴𝑓 𝑀𝐵)) = (𝑧𝐼 ↦ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧)))))
3018, 29, 273eqtr4d 2654 1 (𝜑 → (𝐴𝑓 𝑀(𝐴𝑓 𝑀𝐵)) = 𝐵)
 Colors of variables: wff setvar class 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
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