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Theorem caonncan 6810
Description: Transfer nncan 10161-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 6252 . . . 4 ((𝜑𝑧𝐼) → (𝐴𝑧) ∈ 𝑆)
3 caonncan.b . . . . 5 (𝜑𝐵:𝐼𝑆)
43ffvelrnda 6252 . . . 4 ((𝜑𝑧𝐼) → (𝐵𝑧) ∈ 𝑆)
5 caonncan.z . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
65ralrimivva 2953 . . . . 5 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
76adantr 479 . . . 4 ((𝜑𝑧𝐼) → ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)
8 id 22 . . . . . . 7 (𝑥 = (𝐴𝑧) → 𝑥 = (𝐴𝑧))
9 oveq1 6534 . . . . . . 7 (𝑥 = (𝐴𝑧) → (𝑥𝑀𝑦) = ((𝐴𝑧)𝑀𝑦))
108, 9oveq12d 6545 . . . . . 6 (𝑥 = (𝐴𝑧) → (𝑥𝑀(𝑥𝑀𝑦)) = ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)))
1110eqeq1d 2611 . . . . 5 (𝑥 = (𝐴𝑧) → ((𝑥𝑀(𝑥𝑀𝑦)) = 𝑦 ↔ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = 𝑦))
12 oveq2 6535 . . . . . . 7 (𝑦 = (𝐵𝑧) → ((𝐴𝑧)𝑀𝑦) = ((𝐴𝑧)𝑀(𝐵𝑧)))
1312oveq2d 6543 . . . . . 6 (𝑦 = (𝐵𝑧) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))))
14 id 22 . . . . . 6 (𝑦 = (𝐵𝑧) → 𝑦 = (𝐵𝑧))
1513, 14eqeq12d 2624 . . . . 5 (𝑦 = (𝐵𝑧) → (((𝐴𝑧)𝑀((𝐴𝑧)𝑀𝑦)) = 𝑦 ↔ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧)))
1611, 15rspc2va 3293 . . . 4 ((((𝐴𝑧) ∈ 𝑆 ∧ (𝐵𝑧) ∈ 𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧))
172, 4, 7, 16syl21anc 1316 . . 3 ((𝜑𝑧𝐼) → ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧))) = (𝐵𝑧))
1817mpteq2dva 4666 . 2 (𝜑 → (𝑧𝐼 ↦ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧)))) = (𝑧𝐼 ↦ (𝐵𝑧)))
19 caonncan.i . . 3 (𝜑𝐼𝑉)
20 fvex 6098 . . . 4 (𝐴𝑧) ∈ V
2120a1i 11 . . 3 ((𝜑𝑧𝐼) → (𝐴𝑧) ∈ V)
22 ovex 6555 . . . 4 ((𝐴𝑧)𝑀(𝐵𝑧)) ∈ V
2322a1i 11 . . 3 ((𝜑𝑧𝐼) → ((𝐴𝑧)𝑀(𝐵𝑧)) ∈ V)
241feqmptd 6144 . . 3 (𝜑𝐴 = (𝑧𝐼 ↦ (𝐴𝑧)))
25 fvex 6098 . . . . 5 (𝐵𝑧) ∈ V
2625a1i 11 . . . 4 ((𝜑𝑧𝐼) → (𝐵𝑧) ∈ V)
273feqmptd 6144 . . . 4 (𝜑𝐵 = (𝑧𝐼 ↦ (𝐵𝑧)))
2819, 21, 26, 24, 27offval2 6789 . . 3 (𝜑 → (𝐴𝑓 𝑀𝐵) = (𝑧𝐼 ↦ ((𝐴𝑧)𝑀(𝐵𝑧))))
2919, 21, 23, 24, 28offval2 6789 . 2 (𝜑 → (𝐴𝑓 𝑀(𝐴𝑓 𝑀𝐵)) = (𝑧𝐼 ↦ ((𝐴𝑧)𝑀((𝐴𝑧)𝑀(𝐵𝑧)))))
3018, 29, 273eqtr4d 2653 1 (𝜑 → (𝐴𝑓 𝑀(𝐴𝑓 𝑀𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  cmpt 4637  wf 5786  cfv 5790  (class class class)co 6527  𝑓 cof 6770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772
This theorem is referenced by:  psropprmul  19375
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