Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > caovord3d | Structured version Visualization version 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 5072 | . 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 7360 | . . 3 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ (𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵))) |
8 | caovord3d.5 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑆) | |
9 | 2, 8, 5, 4 | caovordd 7359 | . . 3 ⊢ (𝜑 → (𝐷𝑅𝐵 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵))) |
10 | 7, 9 | bibi12d 348 | . 2 ⊢ (𝜑 → ((𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵) ↔ ((𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))) |
11 | 1, 10 | syl5ibr 248 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 (class class class)co 7159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |