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Theorem caonncan 7663
Description: Transfer nncan 11437-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 7040 . . . 4 ((𝜑𝑧𝐼) → (𝐴𝑧) ∈ 𝑆)
3 caonncan.b . . . . 5 (𝜑𝐵:𝐼𝑆)
43ffvelcdmda 7040 . . . 4 ((𝜑𝑧𝐼) → (𝐵𝑧) ∈ 𝑆)
5 caonncan.z . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
65ralrimivva 3198 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
76adantr 482 . . . 4 ((𝜑𝑧𝐼) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
8 id 22 . . . . . . 7 (𝑥 = (𝐴𝑧) → 𝑥 = (𝐴𝑧))
9 oveq1 7369 . . . . . . 7 (𝑥 = (𝐴𝑧) → (𝑥𝑀𝑦) = ((𝐴𝑧)𝑀𝑦))
108, 9oveq12d 7380 . . . . . 6 (𝑥 = (𝐴𝑧) → (𝑥𝑀(𝑥𝑀𝑦)) = ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)))
1110eqeq1d 2739 . . . . 5 (𝑥 = (𝐴𝑧) → ((𝑥𝑀(𝑥𝑀𝑦)) = 𝑦 ↔ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = 𝑦))
12 oveq2 7370 . . . . . . 7 (𝑦 = (𝐵𝑧) → ((𝐴𝑧)𝑀𝑦) = ((𝐴𝑧)𝑀(𝐵𝑧)))
1312oveq2d 7378 . . . . . 6 (𝑦 = (𝐵𝑧) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))))
14 id 22 . . . . . 6 (𝑦 = (𝐵𝑧) → 𝑦 = (𝐵𝑧))
1513, 14eqeq12d 2753 . . . . 5 (𝑦 = (𝐵𝑧) → (((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = 𝑦 ↔ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧)))
1611, 15rspc2va 3594 . . . 4 ((((𝐴𝑧) ∈ 𝑆 ∧ (𝐵𝑧) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧))
172, 4, 7, 16syl21anc 837 . . 3 ((𝜑𝑧𝐼) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧))
1817mpteq2dva 5210 . 2 (𝜑 → (𝑧𝐼 ↦ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧)))) = (𝑧𝐼 ↦ (𝐵𝑧)))
19 caonncan.i . . 3 (𝜑𝐼𝑉)
20 fvexd 6862 . . 3 ((𝜑𝑧𝐼) → (𝐴𝑧) ∈ V)
21 ovexd 7397 . . 3 ((𝜑𝑧𝐼) → ((𝐴𝑧)𝑀(𝐵𝑧)) ∈ V)
221feqmptd 6915 . . 3 (𝜑𝐴 = (𝑧𝐼 ↦ (𝐴𝑧)))
23 fvexd 6862 . . . 4 ((𝜑𝑧𝐼) → (𝐵𝑧) ∈ V)
243feqmptd 6915 . . . 4 (𝜑𝐵 = (𝑧𝐼 ↦ (𝐵𝑧)))
2519, 20, 23, 22, 24offval2 7642 . . 3 (𝜑 → (𝐴f 𝑀𝐵) = (𝑧𝐼 ↦ ((𝐴𝑧)𝑀(𝐵𝑧))))
2619, 20, 21, 22, 25offval2 7642 . 2 (𝜑 → (𝐴f 𝑀(𝐴f 𝑀𝐵)) = (𝑧𝐼 ↦ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧)))))
2718, 26, 243eqtr4d 2787 1 (𝜑 → (𝐴f 𝑀(𝐴f 𝑀𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065  Vcvv 3448  cmpt 5193  wf 6497  cfv 6501  (class class class)co 7362  f cof 7620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622
This theorem is referenced by:  psropprmul  21625
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