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Mirrors > Home > MPE Home > Th. List > caovordg | Structured version Visualization version GIF version |
Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovordg.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) |
Ref | Expression |
---|---|
caovordg | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovordg.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) | |
2 | 1 | ralrimivvva 2972 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) |
3 | breq1 4656 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦)) | |
4 | oveq2 6658 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑧𝐹𝑥) = (𝑧𝐹𝐴)) | |
5 | 4 | breq1d 4663 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦))) |
6 | 3, 5 | bibi12d 335 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)))) |
7 | breq2 4657 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵)) | |
8 | oveq2 6658 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑧𝐹𝑦) = (𝑧𝐹𝐵)) | |
9 | 8 | breq2d 4665 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))) |
10 | 7, 9 | bibi12d 335 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))) |
11 | oveq1 6657 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧𝐹𝐴) = (𝐶𝐹𝐴)) | |
12 | oveq1 6657 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧𝐹𝐵) = (𝐶𝐹𝐵)) | |
13 | 11, 12 | breq12d 4666 | . . . 4 ⊢ (𝑧 = 𝐶 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵) ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
14 | 13 | bibi2d 332 | . . 3 ⊢ (𝑧 = 𝐶 → ((𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)) ↔ (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))) |
15 | 6, 10, 14 | rspc3v 3325 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))) |
16 | 2, 15 | mpan9 486 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 class class class wbr 4653 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: caovordd 6842 |
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