Theorem mirne 28753

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 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘𝐵) = 𝐴)
2	1	fveq2d 6831	. . . 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 28748	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵)
13	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘(𝑀‘𝐵)) = 𝐵)
14		eqid 2739	. . . . . 6 ⊢ 𝐴 = 𝐴
15	3, 4, 5, 6, 7, 8, 9, 10, 9	mirinv 28752	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 = 𝐴))
16	14, 15	mpbiri 259	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴)
17	16	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → (𝑀‘𝐴) = 𝐴)
18	2, 13, 17	3eqtr3d 2782	. . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → 𝐵 = 𝐴)
19		mirne.1	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
20	19	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → 𝐵 ≠ 𝐴)
21	20	neneqd 2939	. . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐴) → ¬ 𝐵 = 𝐴)
22	18, 21	pm2.65da 822	. 2 ⊢ (𝜑 → ¬ (𝑀‘𝐵) = 𝐴)
23	22	neqned 2941	1 ⊢ (𝜑 → (𝑀‘𝐵) ≠ 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ‘cfv 6485  Basecbs 17170  distcds 17220  TarskiGcstrkg 28513  Itvcitv 28519  LineGclng 28520  pInvGcmir 28738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-trkgc 28534  df-trkgb 28535  df-trkgcb 28536  df-trkg 28539  df-mir 28739

This theorem is referenced by:  mirhl2  28767  sacgr  28917