Theorem colperpexlem2 26519

Description: Lemma for colperpex 26521. 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 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐴 = 𝑄)
3	2	fveq2d 6676	. . . . . . . . 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 2883	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝑀 = 𝐾)
7	6	fveq1d 6674	. . . . . . 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 26450	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶)
17	16	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝑀‘(𝑀‘𝐶)) = 𝐶)
18		colperpexlem.2	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))
19	18	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))
20	7, 17, 19	3eqtr3rd 2867	. . . . . 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 26454	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
24	23	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
25	20, 24	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐵 = 𝐶)
26	25	ex 415	. . . 4 ⊢ (𝜑 → (𝐴 = 𝑄 → 𝐵 = 𝐶))
27	26	necon3ad 3031	. . 3 ⊢ (𝜑 → (𝐵 ≠ 𝐶 → ¬ 𝐴 = 𝑄))
28	1, 27	mpd 15	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝑄)
29	28	neqned 3025	1 ⊢ (𝜑 → 𝐴 ≠ 𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ‘cfv 6357  ⟨“cs3 14206  Basecbs 16485  distcds 16576  TarskiGcstrkg 26218  Itvcitv 26224  LineGclng 26225  pInvGcmir 26440  ∟Gcrag 26481

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-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  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-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  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-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-trkgc 26236  df-trkgb 26237  df-trkgcb 26238  df-trkg 26241  df-mir 26441

This theorem is referenced by:  colperpexlem3  26520