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| Mirrors > Home > MPE Home > Th. List > caovcld | Structured version Visualization version GIF version | ||
| Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) |
| Ref | Expression |
|---|---|
| caovclg.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸) |
| caovcld.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
| caovcld.3 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
| Ref | Expression |
|---|---|
| caovcld | ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (𝜑 → 𝜑) | |
| 2 | caovcld.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
| 3 | caovcld.3 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
| 4 | caovclg.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸) | |
| 5 | 4 | caovclg 7625 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸) |
| 6 | 1, 2, 3, 5 | syl12anc 837 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 (class class class)co 7431 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: caovdir2d 7649 caov4d 7657 climcn2 15629 grpinva 18687 plydivlem1 26335 plydivlem4 26338 |
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