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Mirrors > Home > MPE Home > Th. List > Mathboxes > intopval | Structured version Visualization version GIF version |
Description: The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.) |
Ref | Expression |
---|---|
intopval | ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 intOp 𝑁) = (𝑁 ↑𝑚 (𝑀 × 𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-intop 42345 | . . 3 ⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑𝑚 (𝑚 × 𝑚))) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑𝑚 (𝑚 × 𝑚)))) |
3 | simpr 479 | . . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁) | |
4 | simpl 474 | . . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → 𝑚 = 𝑀) | |
5 | 4 | sqxpeqd 5298 | . . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (𝑚 × 𝑚) = (𝑀 × 𝑀)) |
6 | 3, 5 | oveq12d 6831 | . . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (𝑛 ↑𝑚 (𝑚 × 𝑚)) = (𝑁 ↑𝑚 (𝑀 × 𝑀))) |
7 | 6 | adantl 473 | . 2 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → (𝑛 ↑𝑚 (𝑚 × 𝑚)) = (𝑁 ↑𝑚 (𝑀 × 𝑀))) |
8 | elex 3352 | . . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V) | |
9 | 8 | adantr 472 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑀 ∈ V) |
10 | elex 3352 | . . 3 ⊢ (𝑁 ∈ 𝑊 → 𝑁 ∈ V) | |
11 | 10 | adantl 473 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑁 ∈ V) |
12 | ovexd 6843 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑁 ↑𝑚 (𝑀 × 𝑀)) ∈ V) | |
13 | 2, 7, 9, 11, 12 | ovmpt2d 6953 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 intOp 𝑁) = (𝑁 ↑𝑚 (𝑀 × 𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 × cxp 5264 (class class class)co 6813 ↦ cmpt2 6815 ↑𝑚 cmap 8023 intOp cintop 42342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-iota 6012 df-fun 6051 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-intop 42345 |
This theorem is referenced by: intop 42349 clintopval 42350 |
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