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Theorem caonncan 7734
Description: Transfer nncan 11529-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)
21ffvelcdmda 7099 . . . 4 ((𝜑𝑧𝐼) → (𝐴𝑧) ∈ 𝑆)
3 caonncan.b . . . . 5 (𝜑𝐵:𝐼𝑆)
43ffvelcdmda 7099 . . . 4 ((𝜑𝑧𝐼) → (𝐵𝑧) ∈ 𝑆)
5 caonncan.z . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
65ralrimivva 3198 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
76adantr 479 . . . 4 ((𝜑𝑧𝐼) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
8 id 22 . . . . . . 7 (𝑥 = (𝐴𝑧) → 𝑥 = (𝐴𝑧))
9 oveq1 7433 . . . . . . 7 (𝑥 = (𝐴𝑧) → (𝑥𝑀𝑦) = ((𝐴𝑧)𝑀𝑦))
108, 9oveq12d 7444 . . . . . 6 (𝑥 = (𝐴𝑧) → (𝑥𝑀(𝑥𝑀𝑦)) = ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)))
1110eqeq1d 2730 . . . . 5 (𝑥 = (𝐴𝑧) → ((𝑥𝑀(𝑥𝑀𝑦)) = 𝑦 ↔ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = 𝑦))
12 oveq2 7434 . . . . . . 7 (𝑦 = (𝐵𝑧) → ((𝐴𝑧)𝑀𝑦) = ((𝐴𝑧)𝑀(𝐵𝑧)))
1312oveq2d 7442 . . . . . 6 (𝑦 = (𝐵𝑧) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))))
14 id 22 . . . . . 6 (𝑦 = (𝐵𝑧) → 𝑦 = (𝐵𝑧))
1513, 14eqeq12d 2744 . . . . 5 (𝑦 = (𝐵𝑧) → (((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = 𝑦 ↔ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧)))
1611, 15rspc2va 3623 . . . 4 ((((𝐴𝑧) ∈ 𝑆 ∧ (𝐵𝑧) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧))
172, 4, 7, 16syl21anc 836 . . 3 ((𝜑𝑧𝐼) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧))
1817mpteq2dva 5252 . 2 (𝜑 → (𝑧𝐼 ↦ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧)))) = (𝑧𝐼 ↦ (𝐵𝑧)))
19 caonncan.i . . 3 (𝜑𝐼𝑉)
20 fvexd 6917 . . 3 ((𝜑𝑧𝐼) → (𝐴𝑧) ∈ V)
21 ovexd 7461 . . 3 ((𝜑𝑧𝐼) → ((𝐴𝑧)𝑀(𝐵𝑧)) ∈ V)
221feqmptd 6972 . . 3 (𝜑𝐴 = (𝑧𝐼 ↦ (𝐴𝑧)))
23 fvexd 6917 . . . 4 ((𝜑𝑧𝐼) → (𝐵𝑧) ∈ V)
243feqmptd 6972 . . . 4 (𝜑𝐵 = (𝑧𝐼 ↦ (𝐵𝑧)))
2519, 20, 23, 22, 24offval2 7712 . . 3 (𝜑 → (𝐴f 𝑀𝐵) = (𝑧𝐼 ↦ ((𝐴𝑧)𝑀(𝐵𝑧))))
2619, 20, 21, 22, 25offval2 7712 . 2 (𝜑 → (𝐴f 𝑀(𝐴f 𝑀𝐵)) = (𝑧𝐼 ↦ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧)))))
2718, 26, 243eqtr4d 2778 1 (𝜑 → (𝐴f 𝑀(𝐴f 𝑀𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3058  Vcvv 3473  cmpt 5235  wf 6549  cfv 6553  (class class class)co 7426  f cof 7690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7692
This theorem is referenced by:  psropprmul  22175
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