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Mirrors > Home > MPE Home > Th. List > caovclg | Structured version Visualization version GIF version |
Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.) |
Ref | Expression |
---|---|
caovclg.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸) |
Ref | Expression |
---|---|
caovclg | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovclg.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸) | |
2 | 1 | ralrimivva 3200 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ∈ 𝐸) |
3 | oveq1 7438 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦)) | |
4 | 3 | eleq1d 2824 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) ∈ 𝐸 ↔ (𝐴𝐹𝑦) ∈ 𝐸)) |
5 | oveq2 7439 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵)) | |
6 | 5 | eleq1d 2824 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) ∈ 𝐸 ↔ (𝐴𝐹𝐵) ∈ 𝐸)) |
7 | 4, 6 | rspc2v 3633 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ∈ 𝐸 → (𝐴𝐹𝐵) ∈ 𝐸)) |
8 | 2, 7 | mpan9 506 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 (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 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: caovcld 7626 caovcl 7627 coof 7721 seqcl2 14058 seqcaopr 14077 ercpbl 17596 grpinva 18700 gsumpropd2lem 18705 imasmnd2 18800 imasgrp2 19086 gsumzaddlem 19954 imasrng 20195 imasring 20344 qusrhm 21304 qusmul2idl 21307 mplind 22112 plymullem 26270 fsuppssind 42580 |
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