Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ndmaovcom | Structured version Visualization version Unicode version |
Description: Any operation is commutative outside its domain, analogous to ndmovcom 6821. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
ndmaov.1 |
Ref | Expression |
---|---|
ndmaovcom | (()) (()) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5146 | . . . 4 | |
2 | ndmaov.1 | . . . . . 6 | |
3 | 2 | eqcomi 2631 | . . . . 5 |
4 | 3 | eleq2i 2693 | . . . 4 |
5 | 1, 4 | bitr3i 266 | . . 3 |
6 | ndmaov 41263 | . . 3 (()) | |
7 | 5, 6 | sylnbi 320 | . 2 (()) |
8 | ancom 466 | . . . 4 | |
9 | opelxp 5146 | . . . 4 | |
10 | 3 | eleq2i 2693 | . . . 4 |
11 | 8, 9, 10 | 3bitr2i 288 | . . 3 |
12 | ndmaov 41263 | . . 3 (()) | |
13 | 11, 12 | sylnbi 320 | . 2 (()) |
14 | 7, 13 | eqtr4d 2659 | 1 (()) (()) |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 cop 4183 cxp 5112 cdm 5114 ((caov 41195 |
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 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
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-opab 4713 df-xp 5120 df-fv 5896 df-dfat 41196 df-afv 41197 df-aov 41198 |
This theorem is referenced by: (None) |
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