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Theorem intopval 41609
Description: The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
Assertion
Ref Expression
intopval ((𝑀𝑉𝑁𝑊) → (𝑀 intOp 𝑁) = (𝑁𝑚 (𝑀 × 𝑀)))

Proof of Theorem intopval
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-intop 41606 . . 3 intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛𝑚 (𝑚 × 𝑚)))
21a1i 11 . 2 ((𝑀𝑉𝑁𝑊) → intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛𝑚 (𝑚 × 𝑚))))
3 simpr 475 . . . 4 ((𝑚 = 𝑀𝑛 = 𝑁) → 𝑛 = 𝑁)
4 simpl 471 . . . . 5 ((𝑚 = 𝑀𝑛 = 𝑁) → 𝑚 = 𝑀)
54sqxpeqd 5054 . . . 4 ((𝑚 = 𝑀𝑛 = 𝑁) → (𝑚 × 𝑚) = (𝑀 × 𝑀))
63, 5oveq12d 6544 . . 3 ((𝑚 = 𝑀𝑛 = 𝑁) → (𝑛𝑚 (𝑚 × 𝑚)) = (𝑁𝑚 (𝑀 × 𝑀)))
76adantl 480 . 2 (((𝑀𝑉𝑁𝑊) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (𝑛𝑚 (𝑚 × 𝑚)) = (𝑁𝑚 (𝑀 × 𝑀)))
8 elex 3184 . . 3 (𝑀𝑉𝑀 ∈ V)
98adantr 479 . 2 ((𝑀𝑉𝑁𝑊) → 𝑀 ∈ V)
10 elex 3184 . . 3 (𝑁𝑊𝑁 ∈ V)
1110adantl 480 . 2 ((𝑀𝑉𝑁𝑊) → 𝑁 ∈ V)
12 ovex 6554 . . 3 (𝑁𝑚 (𝑀 × 𝑀)) ∈ V
1312a1i 11 . 2 ((𝑀𝑉𝑁𝑊) → (𝑁𝑚 (𝑀 × 𝑀)) ∈ V)
142, 7, 9, 11, 13ovmpt2d 6663 1 ((𝑀𝑉𝑁𝑊) → (𝑀 intOp 𝑁) = (𝑁𝑚 (𝑀 × 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172   × cxp 5025  (class class class)co 6526  cmpt2 6528  𝑚 cmap 7721   intOp cintop 41603
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-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
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-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  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-br 4578  df-opab 4638  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-iota 5753  df-fun 5791  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-intop 41606
This theorem is referenced by:  intop  41610  clintopval  41611
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