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Theorem colrot1 25810
Description: Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
tglngval.p 𝑃 = (Base‘𝐺)
tglngval.l 𝐿 = (LineG‘𝐺)
tglngval.i 𝐼 = (Itv‘𝐺)
tglngval.g (𝜑𝐺 ∈ TarskiG)
tglngval.x (𝜑𝑋𝑃)
tglngval.y (𝜑𝑌𝑃)
tgcolg.z (𝜑𝑍𝑃)
colrot (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
Assertion
Ref Expression
colrot1 (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))

Proof of Theorem colrot1
StepHypRef Expression
1 colrot . 2 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2 3orrot 1113 . . . 4 ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)))
3 tglngval.p . . . . . 6 𝑃 = (Base‘𝐺)
4 eqid 2799 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
5 tglngval.i . . . . . 6 𝐼 = (Itv‘𝐺)
6 tglngval.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
7 tgcolg.z . . . . . 6 (𝜑𝑍𝑃)
8 tglngval.x . . . . . 6 (𝜑𝑋𝑃)
9 tglngval.y . . . . . 6 (𝜑𝑌𝑃)
103, 4, 5, 6, 7, 8, 9tgbtwncomb 25740 . . . . 5 (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑌𝐼𝑍)))
11 biidd 254 . . . . 5 (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
123, 4, 5, 6, 8, 7, 9tgbtwncomb 25740 . . . . 5 (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑌𝐼𝑋)))
1310, 11, 123orbi123d 1560 . . . 4 (𝜑 → ((𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)) ↔ (𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋))))
142, 13syl5bb 275 . . 3 (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋))))
15 tglngval.l . . . 4 𝐿 = (LineG‘𝐺)
163, 15, 5, 6, 8, 9, 7tgcolg 25805 . . 3 (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
173, 15, 5, 6, 9, 7, 8tgcolg 25805 . . 3 (𝜑 → ((𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ (𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋))))
1814, 16, 173bitr4d 303 . 2 (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)))
191, 18mpbid 224 1 (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 874  w3o 1107   = wceq 1653  wcel 2157  cfv 6101  (class class class)co 6878  Basecbs 16184  distcds 16276  TarskiGcstrkg 25681  Itvcitv 25687  LineGclng 25688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-trkgc 25699  df-trkgb 25700  df-trkgcb 25701  df-trkg 25704
This theorem is referenced by:  colrot2  25811  ncolrot2  25814  ncolncol  25897  midexlem  25943  ragflat3  25957  mideulem2  25982  opphllem  25983  hlpasch  26004  colhp  26018  trgcopy  26052  trgcopyeulem  26053  cgracgr  26066  cgraswap  26068  cgrg3col4  26090  tgasa1  26095
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