Theorem colperpexlem2 28739

Description: Lemma for colperpex 28741. Second part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 10-Nov-2019.)

Hypotheses

Ref	Expression
colperpex.p	⊢ 𝑃 = (Base‘𝐺)
colperpex.d	⊢ − = (dist‘𝐺)
colperpex.i	⊢ 𝐼 = (Itv‘𝐺)
colperpex.l	⊢ 𝐿 = (LineG‘𝐺)
colperpex.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
colperpexlem.s	⊢ 𝑆 = (pInvG‘𝐺)
colperpexlem.m	⊢ 𝑀 = (𝑆‘𝐴)
colperpexlem.n	⊢ 𝑁 = (𝑆‘𝐵)
colperpexlem.k	⊢ 𝐾 = (𝑆‘𝑄)
colperpexlem.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
colperpexlem.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
colperpexlem.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
colperpexlem.q	⊢ (𝜑 → 𝑄 ∈ 𝑃)
colperpexlem.1	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
colperpexlem.2	⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))
colperpexlem2.e	⊢ (𝜑 → 𝐵 ≠ 𝐶)

Assertion

Ref	Expression
colperpexlem2	⊢ (𝜑 → 𝐴 ≠ 𝑄)

Proof of Theorem colperpexlem2

Step	Hyp	Ref	Expression
1		colperpexlem2.e	. . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
2		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐴 = 𝑄)
3	2	fveq2d 6910	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝑆‘𝐴) = (𝑆‘𝑄))
4		colperpexlem.m	. . . . . . . . 9 ⊢ 𝑀 = (𝑆‘𝐴)
5		colperpexlem.k	. . . . . . . . 9 ⊢ 𝐾 = (𝑆‘𝑄)
6	3, 4, 5	3eqtr4g 2802	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝑀 = 𝐾)
7	6	fveq1d 6908	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝑀‘(𝑀‘𝐶)) = (𝐾‘(𝑀‘𝐶)))
8		colperpex.p	. . . . . . . . 9 ⊢ 𝑃 = (Base‘𝐺)
9		colperpex.d	. . . . . . . . 9 ⊢ − = (dist‘𝐺)
10		colperpex.i	. . . . . . . . 9 ⊢ 𝐼 = (Itv‘𝐺)
11		colperpex.l	. . . . . . . . 9 ⊢ 𝐿 = (LineG‘𝐺)
12		colperpexlem.s	. . . . . . . . 9 ⊢ 𝑆 = (pInvG‘𝐺)
13		colperpex.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
14		colperpexlem.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
15		colperpexlem.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
16	8, 9, 10, 11, 12, 13, 14, 4, 15	mirmir 28670	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶)
17	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝑀‘(𝑀‘𝐶)) = 𝐶)
18		colperpexlem.2	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))
19	18	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))
20	7, 17, 19	3eqtr3rd 2786	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝑁‘𝐶) = 𝐶)
21		colperpexlem.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
22		colperpexlem.n	. . . . . . . 8 ⊢ 𝑁 = (𝑆‘𝐵)
23	8, 9, 10, 11, 12, 13, 21, 22, 15	mirinv 28674	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
24	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
25	20, 24	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐵 = 𝐶)
26	25	ex 412	. . . 4 ⊢ (𝜑 → (𝐴 = 𝑄 → 𝐵 = 𝐶))
27	26	necon3ad 2953	. . 3 ⊢ (𝜑 → (𝐵 ≠ 𝐶 → ¬ 𝐴 = 𝑄))
28	1, 27	mpd 15	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝑄)
29	28	neqned 2947	1 ⊢ (𝜑 → 𝐴 ≠ 𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ‘cfv 6561  ⟨“cs3 14881  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442  pInvGcmir 28660  ∟Gcrag 28701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461  df-mir 28661

This theorem is referenced by:  colperpexlem3  28740