Theorem colperpexlem2 29063

Description: Lemma for colperpex 29065. 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 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐴 = 𝑄)
3	2	fveq2d 6889	. . . . . . . . 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 2825	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝑀 = 𝐾)
7	6	fveq1d 6887	. . . . . . 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 28990	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶)
17	16	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝑀‘(𝑀‘𝐶)) = 𝐶)
18		colperpexlem.2	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))
19	18	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶))
20	7, 17, 19	3eqtr3rd 2809	. . . . . 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 28994	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
24	23	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
25	20, 24	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐵 = 𝐶)
26	25	ex 418	. . . 4 ⊢ (𝜑 → (𝐴 = 𝑄 → 𝐵 = 𝐶))
27	26	necon3ad 2973	. . 3 ⊢ (𝜑 → (𝐵 ≠ 𝐶 → ¬ 𝐴 = 𝑄))
28	1, 27	mpd 16	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝑄)
29	28	neqned 2967	1 ⊢ (𝜑 → 𝐴 ≠ 𝑄)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ‘cfv 6540  ⟨“cs3 14903  Basecbs 17291  distcds 17341  TarskiGcstrkg 28747  Itvcitv 28753  LineGclng 28754  pInvGcmir 28980  ∟Gcrag 29024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-trkgc 28768  df-trkgb 28769  df-trkgcb 28770  df-trkg 28773  df-mir 28981

This theorem is used by:  colperpexlem3  29064