Theorem colperpexlem2 28665

Description: Lemma for colperpex 28667. 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 6865	. . . . . . . . 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 2790	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝑀 = 𝐾)
7	6	fveq1d 6863	. . . . . . 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 28596	. . . . . . . 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 2774	. . . . . 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 28600	. . . . . . 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 2939	. . 3 ⊢ (𝜑 → (𝐵 ≠ 𝐶 → ¬ 𝐴 = 𝑄))
28	1, 27	mpd 15	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝑄)
29	28	neqned 2933	1 ⊢ (𝜑 → 𝐴 ≠ 𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ‘cfv 6514  ⟨“cs3 14815  Basecbs 17186  distcds 17236  TarskiGcstrkg 28361  Itvcitv 28367  LineGclng 28368  pInvGcmir 28586  ∟Gcrag 28627

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-trkgc 28382  df-trkgb 28383  df-trkgcb 28384  df-trkg 28387  df-mir 28587

This theorem is referenced by:  colperpexlem3  28666