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Mirrors > Home > ILE Home > Th. List > caovord3d | GIF version |
Description: Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovordg.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) |
caovordd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
caovordd.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
caovordd.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
caovord2d.com | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
caovord3d.5 | ⊢ (𝜑 → 𝐷 ∈ 𝑆) |
Ref | Expression |
---|---|
caovord3d | ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3940 | . 2 ⊢ ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → ((𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵))) | |
2 | caovordg.1 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) | |
3 | caovordd.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | caovordd.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
5 | caovordd.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
6 | caovord2d.com | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) | |
7 | 2, 3, 4, 5, 6 | caovord2d 5948 | . . 3 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ (𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵))) |
8 | caovord3d.5 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑆) | |
9 | 2, 8, 5, 4 | caovordd 5947 | . . 3 ⊢ (𝜑 → (𝐷𝑅𝐵 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵))) |
10 | 7, 9 | bibi12d 234 | . 2 ⊢ (𝜑 → ((𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵) ↔ ((𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))) |
11 | 1, 10 | syl5ibr 155 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 |
This theorem is referenced by: ordpipqqs 7206 ltsrprg 7579 |
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