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Mirrors > Home > MPE Home > Th. List > colcom | Structured version Visualization version Unicode version |
Description: Swapping the points defining a line keeps it unchanged. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.) |
Ref | Expression |
---|---|
tglngval.p | |
tglngval.l | LineG |
tglngval.i | Itv |
tglngval.g | TarskiG |
tglngval.x | |
tglngval.y | |
tgcolg.z | |
colrot |
Ref | Expression |
---|---|
colcom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | colrot | . 2 | |
2 | 3orcomb 1048 | . . . 4 | |
3 | tglngval.p | . . . . . 6 | |
4 | eqid 2622 | . . . . . 6 | |
5 | tglngval.i | . . . . . 6 Itv | |
6 | tglngval.g | . . . . . 6 TarskiG | |
7 | tglngval.x | . . . . . 6 | |
8 | tgcolg.z | . . . . . 6 | |
9 | tglngval.y | . . . . . 6 | |
10 | 3, 4, 5, 6, 7, 8, 9 | tgbtwncomb 25384 | . . . . 5 |
11 | 3, 4, 5, 6, 7, 9, 8 | tgbtwncomb 25384 | . . . . 5 |
12 | 3, 4, 5, 6, 8, 7, 9 | tgbtwncomb 25384 | . . . . 5 |
13 | 10, 11, 12 | 3orbi123d 1398 | . . . 4 |
14 | 2, 13 | syl5bb 272 | . . 3 |
15 | tglngval.l | . . . 4 LineG | |
16 | 3, 15, 5, 6, 7, 9, 8 | tgcolg 25449 | . . 3 |
17 | 3, 15, 5, 6, 9, 7, 8 | tgcolg 25449 | . . 3 |
18 | 14, 16, 17 | 3bitr4d 300 | . 2 |
19 | 1, 18 | mpbid 222 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wo 383 w3o 1036 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 cds 15950 TarskiGcstrkg 25329 Itvcitv 25335 LineGclng 25336 |
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-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 |
This theorem is referenced by: ncolcom 25456 tglineeltr 25526 mirtrcgr 25578 symquadlem 25584 midexlem 25587 colperpexlem1 25622 mideulem2 25626 opphllem 25627 hlpasch 25648 colhp 25662 trgcopy 25696 cgrg3col4 25734 tgasa1 25739 |
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