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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 485 | . . 3 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀) | |
2 | oveq 7151 | . . . . . 6 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦)) | |
3 | oveq 7151 | . . . . . 6 ⊢ (𝑜 = ⚬ → (𝑦𝑜𝑥) = (𝑦 ⚬ 𝑥)) | |
4 | 2, 3 | eqeq12d 2834 | . . . . 5 ⊢ (𝑜 = ⚬ → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥))) |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥))) |
6 | 1, 5 | raleqbidv 3399 | . . 3 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥))) |
7 | 1, 6 | raleqbidv 3399 | . 2 ⊢ ((𝑜 = ⚬ ∧ 𝑚 = 𝑀) → (∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥))) |
8 | df-comlaw 44022 | . 2 ⊢ comLaw = {〈𝑜, 𝑚〉 ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)} | |
9 | 7, 8 | brabga 5412 | 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 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 class class class wbr 5057 (class class class)co 7145 comLaw ccomlaw 44020 |
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-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-iota 6307 df-fv 6356 df-ov 7148 df-comlaw 44022 |
This theorem is referenced by: (None) |
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