Theorem colperpexlem2 27090

Description: Lemma for colperpex 27092. 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 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐴 = 𝑄)
3	2	fveq2d 6775	. . . . . . . . 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 2805	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝑀 = 𝐾)
7	6	fveq1d 6773	. . . . . . 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 27021	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶)
17	16	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝑀‘(𝑀‘𝐶)) = 𝐶)
18		colperpexlem.2	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))
19	18	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))
20	7, 17, 19	3eqtr3rd 2789	. . . . . 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 27025	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
24	23	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
25	20, 24	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐵 = 𝐶)
26	25	ex 413	. . . 4 ⊢ (𝜑 → (𝐴 = 𝑄 → 𝐵 = 𝐶))
27	26	necon3ad 2958	. . 3 ⊢ (𝜑 → (𝐵 ≠ 𝐶 → ¬ 𝐴 = 𝑄))
28	1, 27	mpd 15	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝑄)
29	28	neqned 2952	1 ⊢ (𝜑 → 𝐴 ≠ 𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110   ≠ wne 2945  ‘cfv 6432  ⟨“cs3 14553  Basecbs 16910  distcds 16969  TarskiGcstrkg 26786  Itvcitv 26792  LineGclng 26793  pInvGcmir 27011  ∟Gcrag 27052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-trkgc 26807  df-trkgb 26808  df-trkgcb 26809  df-trkg 26812  df-mir 27012

This theorem is referenced by:  colperpexlem3  27091