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Mirrors > Home > ILE Home > Th. List > caovcld | 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 19 | . 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 6026 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸) |
6 | 1, 2, 3, 5 | syl12anc 1236 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 (class class class)co 5874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-iota 5178 df-fv 5224 df-ov 5877 |
This theorem is referenced by: caovdir2d 6050 caov4d 6058 caovdilemd 6065 caovlem2d 6066 ecopovtrn 6631 ecopovtrng 6634 ordpipqqs 7372 ltanqg 7398 ltmnqg 7399 recexprlem1ssu 7632 mulgt0sr 7776 mulextsr1lem 7778 axmulass 7871 frec2uzrdg 10408 frecuzrdgsuc 10413 frecuzrdgsuctlem 10422 iseqovex 10455 seq3val 10457 seqf 10460 seq3p1 10461 seqp1cd 10465 seq3clss 10466 seq3distr 10512 climcn2 11316 qusaddvallemg 12751 grprinvd 12804 |
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