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| Description: Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.) | 
| Ref | Expression | 
|---|---|
| tglngval.p | ⊢ 𝑃 = (Base‘𝐺) | 
| tglngval.l | ⊢ 𝐿 = (LineG‘𝐺) | 
| tglngval.i | ⊢ 𝐼 = (Itv‘𝐺) | 
| tglngval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) | 
| tglngval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) | 
| tglngval.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) | 
| tgcolg.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) | 
| colrot | ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) | 
| Ref | Expression | 
|---|---|
| colrot2 | ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tglngval.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
| 2 | tglngval.l | . 2 ⊢ 𝐿 = (LineG‘𝐺) | |
| 3 | tglngval.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
| 4 | tglngval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 5 | tglngval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
| 6 | tgcolg.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
| 7 | tglngval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
| 8 | colrot | . . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) | |
| 9 | 1, 2, 3, 4, 7, 5, 6, 8 | colrot1 28568 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) | 
| 10 | 1, 2, 3, 4, 5, 6, 7, 9 | colrot1 28568 | 1 ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 TarskiGcstrkg 28436 Itvcitv 28442 LineGclng 28443 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-trkgc 28457 df-trkgb 28458 df-trkgcb 28459 df-trkg 28462 | 
| This theorem is referenced by: ncolrot1 28571 tglineeltr 28640 ncolncol 28655 symquadlem 28698 hlpasch 28765 hphl 28780 trgcopy 28813 | 
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