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Theorem intopval 42699
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 42696 . . 3 intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛𝑚 (𝑚 × 𝑚)))
21a1i 11 . 2 ((𝑀𝑉𝑁𝑊) → intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛𝑚 (𝑚 × 𝑚))))
3 simpr 479 . . . 4 ((𝑚 = 𝑀𝑛 = 𝑁) → 𝑛 = 𝑁)
4 simpl 476 . . . . 5 ((𝑚 = 𝑀𝑛 = 𝑁) → 𝑚 = 𝑀)
54sqxpeqd 5378 . . . 4 ((𝑚 = 𝑀𝑛 = 𝑁) → (𝑚 × 𝑚) = (𝑀 × 𝑀))
63, 5oveq12d 6928 . . 3 ((𝑚 = 𝑀𝑛 = 𝑁) → (𝑛𝑚 (𝑚 × 𝑚)) = (𝑁𝑚 (𝑀 × 𝑀)))
76adantl 475 . 2 (((𝑀𝑉𝑁𝑊) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (𝑛𝑚 (𝑚 × 𝑚)) = (𝑁𝑚 (𝑀 × 𝑀)))
8 elex 3429 . . 3 (𝑀𝑉𝑀 ∈ V)
98adantr 474 . 2 ((𝑀𝑉𝑁𝑊) → 𝑀 ∈ V)
10 elex 3429 . . 3 (𝑁𝑊𝑁 ∈ V)
1110adantl 475 . 2 ((𝑀𝑉𝑁𝑊) → 𝑁 ∈ V)
12 ovexd 6944 . 2 ((𝑀𝑉𝑁𝑊) → (𝑁𝑚 (𝑀 × 𝑀)) ∈ V)
132, 7, 9, 11, 12ovmpt2d 7053 1 ((𝑀𝑉𝑁𝑊) → (𝑀 intOp 𝑁) = (𝑁𝑚 (𝑀 × 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  Vcvv 3414   × cxp 5344  (class class class)co 6910  cmpt2 6912  𝑚 cmap 8127   intOp cintop 42693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-intop 42696
This theorem is referenced by:  intop  42700  clintopval  42701
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