Theorem mirne 28676

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 6909	. . . 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 28671	. . . . 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 28675	. . . . . 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 2784	. . 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 2944	. . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → ¬ 𝐵 = 𝐴)
22	18, 21	pm2.65da 816	. 2 ⊢ (𝜑 → ¬ (𝑀‘𝐵) = 𝐴)
23	22	neqned 2946	1 ⊢ (𝜑 → (𝑀‘𝐵) ≠ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ‘cfv 6560  Basecbs 17248  distcds 17307  TarskiGcstrkg 28436  Itvcitv 28442  LineGclng 28443  pInvGcmir 28661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-trkgc 28457  df-trkgb 28458  df-trkgcb 28459  df-trkg 28462  df-mir 28662

This theorem is referenced by:  mirhl2  28690  sacgr  28840