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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 6716 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) |
3 | ancom 465 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ↔ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) | |
4 | 1 | ndmov 6716 | . . 3 ⊢ (¬ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵𝐹𝐴) = ∅) |
5 | 3, 4 | sylnbi 319 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐵𝐹𝐴) = ∅) |
6 | 2, 5 | eqtr4d 2647 | 1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∅c0 3874 × cxp 5036 dom cdm 5038 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: addcompi 9595 mulcompi 9597 addcompq 9651 addcomnq 9652 mulcompq 9653 mulcomnq 9654 addcompr 9722 mulcompr 9724 addcomsr 9787 mulcomsr 9789 addcomgi 37681 |
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