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Theorem intopval 44037
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 44034 . . 3 intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛m (𝑚 × 𝑚)))
21a1i 11 . 2 ((𝑀𝑉𝑁𝑊) → intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛m (𝑚 × 𝑚))))
3 simpr 485 . . . 4 ((𝑚 = 𝑀𝑛 = 𝑁) → 𝑛 = 𝑁)
4 simpl 483 . . . . 5 ((𝑚 = 𝑀𝑛 = 𝑁) → 𝑚 = 𝑀)
54sqxpeqd 5580 . . . 4 ((𝑚 = 𝑀𝑛 = 𝑁) → (𝑚 × 𝑚) = (𝑀 × 𝑀))
63, 5oveq12d 7163 . . 3 ((𝑚 = 𝑀𝑛 = 𝑁) → (𝑛m (𝑚 × 𝑚)) = (𝑁m (𝑀 × 𝑀)))
76adantl 482 . 2 (((𝑀𝑉𝑁𝑊) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (𝑛m (𝑚 × 𝑚)) = (𝑁m (𝑀 × 𝑀)))
8 elex 3510 . . 3 (𝑀𝑉𝑀 ∈ V)
98adantr 481 . 2 ((𝑀𝑉𝑁𝑊) → 𝑀 ∈ V)
10 elex 3510 . . 3 (𝑁𝑊𝑁 ∈ V)
1110adantl 482 . 2 ((𝑀𝑉𝑁𝑊) → 𝑁 ∈ V)
12 ovexd 7180 . 2 ((𝑀𝑉𝑁𝑊) → (𝑁m (𝑀 × 𝑀)) ∈ V)
132, 7, 9, 11, 12ovmpod 7291 1 ((𝑀𝑉𝑁𝑊) → (𝑀 intOp 𝑁) = (𝑁m (𝑀 × 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492   × cxp 5546  (class class class)co 7145  cmpo 7147  m cmap 8395   intOp cintop 44031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-intop 44034
This theorem is referenced by:  intop  44038  clintopval  44039
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