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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscomlaw | Structured version Visualization version GIF version |
Description: The predicate "is a commutative operation". (Contributed by AV, 20-Jan-2020.) |
Ref | Expression |
---|---|
iscomlaw | ⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ comLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 486 | . . 3 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀) | |
2 | oveq 7364 | . . . . . 6 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦)) | |
3 | oveq 7364 | . . . . . 6 ⊢ (𝑜 = ⚬ → (𝑦𝑜𝑥) = (𝑦 ⚬ 𝑥)) | |
4 | 2, 3 | eqeq12d 2749 | . . . . 5 ⊢ (𝑜 = ⚬ → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥))) |
5 | 4 | adantr 482 | . . . 4 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥))) |
6 | 1, 5 | raleqbidv 3318 | . . 3 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥))) |
7 | 1, 6 | raleqbidv 3318 | . 2 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥))) |
8 | df-comlaw 46207 | . 2 ⊢ comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)} | |
9 | 7, 8 | brabga 5492 | 1 ⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ comLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 class class class wbr 5106 (class class class)co 7358 comLaw ccomlaw 46205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-iota 6449 df-fv 6505 df-ov 7361 df-comlaw 46207 |
This theorem is referenced by: (None) |
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