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Theorem intopval 44415
 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 44412 . . 3 intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛m (𝑚 × 𝑚)))
21a1i 11 . 2 ((𝑀𝑉𝑁𝑊) → intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛m (𝑚 × 𝑚))))
3 simpr 488 . . . 4 ((𝑚 = 𝑀𝑛 = 𝑁) → 𝑛 = 𝑁)
4 simpl 486 . . . . 5 ((𝑚 = 𝑀𝑛 = 𝑁) → 𝑚 = 𝑀)
54sqxpeqd 5575 . . . 4 ((𝑚 = 𝑀𝑛 = 𝑁) → (𝑚 × 𝑚) = (𝑀 × 𝑀))
63, 5oveq12d 7169 . . 3 ((𝑚 = 𝑀𝑛 = 𝑁) → (𝑛m (𝑚 × 𝑚)) = (𝑁m (𝑀 × 𝑀)))
76adantl 485 . 2 (((𝑀𝑉𝑁𝑊) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (𝑛m (𝑚 × 𝑚)) = (𝑁m (𝑀 × 𝑀)))
8 elex 3498 . . 3 (𝑀𝑉𝑀 ∈ V)
98adantr 484 . 2 ((𝑀𝑉𝑁𝑊) → 𝑀 ∈ V)
10 elex 3498 . . 3 (𝑁𝑊𝑁 ∈ V)
1110adantl 485 . 2 ((𝑀𝑉𝑁𝑊) → 𝑁 ∈ V)
12 ovexd 7186 . 2 ((𝑀𝑉𝑁𝑊) → (𝑁m (𝑀 × 𝑀)) ∈ V)
132, 7, 9, 11, 12ovmpod 7297 1 ((𝑀𝑉𝑁𝑊) → (𝑀 intOp 𝑁) = (𝑁m (𝑀 × 𝑀)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  Vcvv 3480   × cxp 5541  (class class class)co 7151   ∈ cmpo 7153   ↑m cmap 8404   intOp cintop 44409 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6304  df-fun 6347  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-intop 44412 This theorem is referenced by:  intop  44416  clintopval  44417
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