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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 5891 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸) |
6 | 1, 2, 3, 5 | syl12anc 1199 | 1 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1465 (class class class)co 5742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 |
This theorem is referenced by: caovdir2d 5915 caov4d 5923 caovdilemd 5930 caovlem2d 5931 grprinvd 5934 ecopovtrn 6494 ecopovtrng 6497 ordpipqqs 7150 ltanqg 7176 ltmnqg 7177 recexprlem1ssu 7410 mulgt0sr 7554 mulextsr1lem 7556 axmulass 7649 frec2uzrdg 10150 frecuzrdgsuc 10155 frecuzrdgsuctlem 10164 iseqovex 10197 seq3val 10199 seqf 10202 seq3p1 10203 seqp1cd 10207 seq3clss 10208 seq3distr 10254 climcn2 11046 |
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