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Mirrors > Home > MPE Home > Th. List > ndmovcom | Structured version Visualization version GIF version |
Description: Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.) |
Ref | Expression |
---|---|
ndmov.1 | ⊢ dom 𝐹 = (𝑆 × 𝑆) |
Ref | Expression |
---|---|
ndmovcom | ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndmov.1 | . . 3 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
2 | 1 | ndmov 7617 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
3 | ancom 460 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ↔ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) | |
4 | 1 | ndmov 7617 | . . 3 ⊢ (¬ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵𝐹𝐴) = ∅) |
5 | 3, 4 | sylnbi 330 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵𝐹𝐴) = ∅) |
6 | 2, 5 | eqtr4d 2778 | 1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∅c0 4339 × cxp 5687 dom cdm 5689 (class class class)co 7431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-dm 5699 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: addcompi 10932 mulcompi 10934 addcompq 10988 addcomnq 10989 mulcompq 10990 mulcomnq 10991 addcompr 11059 mulcompr 11061 addcomsr 11125 mulcomsr 11127 addcomgi 44452 |
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