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Mirrors > Home > MPE Home > Th. List > caovord | Structured version Visualization version GIF version |
Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) |
Ref | Expression |
---|---|
caovord.1 | ⊢ 𝐴 ∈ V |
caovord.2 | ⊢ 𝐵 ∈ V |
caovord.3 | ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) |
Ref | Expression |
---|---|
caovord | ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6556 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝐹𝐴) = (𝐶𝐹𝐴)) | |
2 | oveq1 6556 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝐹𝐵) = (𝐶𝐹𝐵)) | |
3 | 1, 2 | breq12d 4596 | . . 3 ⊢ (𝑧 = 𝐶 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵) ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
4 | 3 | bibi2d 331 | . 2 ⊢ (𝑧 = 𝐶 → ((𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)) ↔ (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))) |
5 | caovord.1 | . . 3 ⊢ 𝐴 ∈ V | |
6 | caovord.2 | . . 3 ⊢ 𝐵 ∈ V | |
7 | breq1 4586 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦)) | |
8 | oveq2 6557 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧𝐹𝑥) = (𝑧𝐹𝐴)) | |
9 | 8 | breq1d 4593 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦))) |
10 | 7, 9 | bibi12d 334 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)))) |
11 | breq2 4587 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵)) | |
12 | oveq2 6557 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑧𝐹𝑦) = (𝑧𝐹𝐵)) | |
13 | 12 | breq2d 4595 | . . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))) |
14 | 11, 13 | bibi12d 334 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))) |
15 | 10, 14 | sylan9bb 732 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))) |
16 | 15 | imbi2d 329 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) ↔ (𝑧 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))))) |
17 | caovord.3 | . . 3 ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) | |
18 | 5, 6, 16, 17 | vtocl2 3234 | . 2 ⊢ (𝑧 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))) |
19 | 4, 18 | vtoclga 3245 | 1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: caovord2 6744 caovord3 6745 genpcl 9709 |
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