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Mirrors > Home > MPE Home > Th. List > caovord2 | Structured version Visualization version GIF version |
Description: Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.) |
Ref | Expression |
---|---|
caovord.1 | ⊢ 𝐴 ∈ V |
caovord.2 | ⊢ 𝐵 ∈ V |
caovord.3 | ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) |
caovord2.3 | ⊢ 𝐶 ∈ V |
caovord2.com | ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) |
Ref | Expression |
---|---|
caovord2 | ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovord.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | caovord.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | caovord.3 | . . 3 ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) | |
4 | 1, 2, 3 | caovord 6845 | . 2 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
5 | caovord2.3 | . . . 4 ⊢ 𝐶 ∈ V | |
6 | caovord2.com | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
7 | 5, 1, 6 | caovcom 6831 | . . 3 ⊢ (𝐶𝐹𝐴) = (𝐴𝐹𝐶) |
8 | 5, 2, 6 | caovcom 6831 | . . 3 ⊢ (𝐶𝐹𝐵) = (𝐵𝐹𝐶) |
9 | 7, 8 | breq12i 4662 | . 2 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)) |
10 | 4, 9 | syl6bb 276 | 1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 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: caovord3 6847 genpnmax 9829 addclprlem1 9838 mulclprlem 9841 distrlem4pr 9848 ltexprlem6 9863 reclem3pr 9871 ltsosr 9915 |
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