Theorem mirne 28735

Description: Mirror of non-center point cannot be the center point. (Contributed by Thierry Arnoux, 27-Sep-2020.)

Hypotheses

Ref	Expression
mirval.p	⊢ 𝑃 = (Base‘𝐺)
mirval.d	⊢ − = (dist‘𝐺)
mirval.i	⊢ 𝐼 = (Itv‘𝐺)
mirval.l	⊢ 𝐿 = (LineG‘𝐺)
mirval.s	⊢ 𝑆 = (pInvG‘𝐺)
mirval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mirval.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirfv.m	⊢ 𝑀 = (𝑆‘𝐴)
mirinv.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
mirne.1	⊢ (𝜑 → 𝐵 ≠ 𝐴)

Assertion

Ref	Expression
mirne	⊢ (𝜑 → (𝑀‘𝐵) ≠ 𝐴)

Proof of Theorem mirne

Step	Hyp	Ref	Expression
1		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘𝐵) = 𝐴)
2	1	fveq2d 6844	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘(𝑀‘𝐵)) = (𝑀‘𝐴))
3		mirval.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
4		mirval.d	. . . . . 6 ⊢ − = (dist‘𝐺)
5		mirval.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
6		mirval.l	. . . . . 6 ⊢ 𝐿 = (LineG‘𝐺)
7		mirval.s	. . . . . 6 ⊢ 𝑆 = (pInvG‘𝐺)
8		mirval.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9		mirval.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
10		mirfv.m	. . . . . 6 ⊢ 𝑀 = (𝑆‘𝐴)
11		mirinv.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
12	3, 4, 5, 6, 7, 8, 9, 10, 11	mirmir 28730	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵)
13	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘(𝑀‘𝐵)) = 𝐵)
14		eqid 2736	. . . . . 6 ⊢ 𝐴 = 𝐴
15	3, 4, 5, 6, 7, 8, 9, 10, 9	mirinv 28734	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 = 𝐴))
16	14, 15	mpbiri 258	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴)
17	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘𝐴) = 𝐴)
18	2, 13, 17	3eqtr3d 2779	. . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → 𝐵 = 𝐴)
19		mirne.1	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
20	19	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → 𝐵 ≠ 𝐴)
21	20	neneqd 2937	. . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → ¬ 𝐵 = 𝐴)
22	18, 21	pm2.65da 817	. 2 ⊢ (𝜑 → ¬ (𝑀‘𝐵) = 𝐴)
23	22	neqned 2939	1 ⊢ (𝜑 → (𝑀‘𝐵) ≠ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ‘cfv 6498  Basecbs 17179  distcds 17229  TarskiGcstrkg 28495  Itvcitv 28501  LineGclng 28502  pInvGcmir 28720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-trkgc 28516  df-trkgb 28517  df-trkgcb 28518  df-trkg 28521  df-mir 28721

This theorem is referenced by:  mirhl2  28749  sacgr  28899