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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 7543 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸) |
6 | 1, 2, 3, 5 | syl12anc 835 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 (class class class)co 7354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3064 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-iota 6446 df-fv 6502 df-ov 7357 |
This theorem is referenced by: caovdir2d 7567 caov4d 7575 climcn2 15472 grprinvd 18526 plydivlem1 25649 plydivlem4 25652 |
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