MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caonncan Structured version   Visualization version   GIF version

Theorem caonncan 6977
Description: Transfer nncan 10348-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 6399 . . . 4 ((𝜑𝑧𝐼) → (𝐴𝑧) ∈ 𝑆)
3 caonncan.b . . . . 5 (𝜑𝐵:𝐼𝑆)
43ffvelrnda 6399 . . . 4 ((𝜑𝑧𝐼) → (𝐵𝑧) ∈ 𝑆)
5 caonncan.z . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
65ralrimivva 3000 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
76adantr 480 . . . 4 ((𝜑𝑧𝐼) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
8 id 22 . . . . . . 7 (𝑥 = (𝐴𝑧) → 𝑥 = (𝐴𝑧))
9 oveq1 6697 . . . . . . 7 (𝑥 = (𝐴𝑧) → (𝑥𝑀𝑦) = ((𝐴𝑧)𝑀𝑦))
108, 9oveq12d 6708 . . . . . 6 (𝑥 = (𝐴𝑧) → (𝑥𝑀(𝑥𝑀𝑦)) = ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)))
1110eqeq1d 2653 . . . . 5 (𝑥 = (𝐴𝑧) → ((𝑥𝑀(𝑥𝑀𝑦)) = 𝑦 ↔ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = 𝑦))
12 oveq2 6698 . . . . . . 7 (𝑦 = (𝐵𝑧) → ((𝐴𝑧)𝑀𝑦) = ((𝐴𝑧)𝑀(𝐵𝑧)))
1312oveq2d 6706 . . . . . 6 (𝑦 = (𝐵𝑧) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))))
14 id 22 . . . . . 6 (𝑦 = (𝐵𝑧) → 𝑦 = (𝐵𝑧))
1513, 14eqeq12d 2666 . . . . 5 (𝑦 = (𝐵𝑧) → (((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = 𝑦 ↔ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧)))
1611, 15rspc2va 3354 . . . 4 ((((𝐴𝑧) ∈ 𝑆 ∧ (𝐵𝑧) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧))
172, 4, 7, 16syl21anc 1365 . . 3 ((𝜑𝑧𝐼) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧))
1817mpteq2dva 4777 . 2 (𝜑 → (𝑧𝐼 ↦ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧)))) = (𝑧𝐼 ↦ (𝐵𝑧)))
19 caonncan.i . . 3 (𝜑𝐼𝑉)
20 fvexd 6241 . . 3 ((𝜑𝑧𝐼) → (𝐴𝑧) ∈ V)
21 ovexd 6720 . . 3 ((𝜑𝑧𝐼) → ((𝐴𝑧)𝑀(𝐵𝑧)) ∈ V)
221feqmptd 6288 . . 3 (𝜑𝐴 = (𝑧𝐼 ↦ (𝐴𝑧)))
23 fvexd 6241 . . . 4 ((𝜑𝑧𝐼) → (𝐵𝑧) ∈ V)
243feqmptd 6288 . . . 4 (𝜑𝐵 = (𝑧𝐼 ↦ (𝐵𝑧)))
2519, 20, 23, 22, 24offval2 6956 . . 3 (𝜑 → (𝐴𝑓 𝑀𝐵) = (𝑧𝐼 ↦ ((𝐴𝑧)𝑀(𝐵𝑧))))
2619, 20, 21, 22, 25offval2 6956 . 2 (𝜑 → (𝐴𝑓 𝑀(𝐴𝑓 𝑀𝐵)) = (𝑧𝐼 ↦ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧)))))
2718, 26, 243eqtr4d 2695 1 (𝜑 → (𝐴𝑓 𝑀(𝐴𝑓 𝑀𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939
This theorem is referenced by:  psropprmul  19656
  Copyright terms: Public domain W3C validator