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Mirrors > Home > MPE Home > Th. List > colrot2 | Structured version Visualization version GIF version |
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 26347 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) |
10 | 1, 2, 3, 4, 5, 6, 7, 9 | colrot1 26347 | 1 ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 TarskiGcstrkg 26218 Itvcitv 26224 LineGclng 26225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-trkgc 26236 df-trkgb 26237 df-trkgcb 26238 df-trkg 26241 |
This theorem is referenced by: ncolrot1 26350 tglineeltr 26419 ncolncol 26434 symquadlem 26477 hlpasch 26544 hphl 26559 trgcopy 26592 |
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