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Theorem mircinv 28982

Description: The center point is invariant of a point inversion. (Contributed by Thierry Arnoux, 25-Aug-2019.)

**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)

**Assertion**
Ref	Expression
mircinv	⊢ (𝜑 → (𝑀‘𝐴) = 𝐴)

**Proof of Theorem mircinv**
Step	Hyp	Ref	Expression
1		eqid 2766	. 2 ⊢ 𝐴 = 𝐴
2		mirval.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
3		mirval.d	. . 3 ⊢ − = (dist‘𝐺)
4		mirval.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
5		mirval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
6		mirval.s	. . 3 ⊢ 𝑆 = (pInvG‘𝐺)
7		mirval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8		mirval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9		mirfv.m	. . 3 ⊢ 𝑀 = (𝑆‘𝐴)
10	2, 3, 4, 5, 6, 7, 8, 9, 8	mirinv 28980	. 2 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 = 𝐴))
11	1, 10	mpbiri 261	1 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 ‘cfv 6543 Basecbs 17294 distcds 17344 TarskiGcstrkg 28733 Itvcitv 28739 LineGclng 28740 pInvGcmir 28966

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 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pr 5409

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-trkgc 28754 df-trkgb 28755 df-trkgcb 28756 df-trkg 28759 df-mir 28967

This theorem is used by: mirln 28990 mirconn 28992 mirbtwnhl 28994 midexlem 29006 ragtrivb 29019 colperpexlem1 29048 colperpexlem3 29050 midex 29055 mirplncl 29114 lmieu 29130 prlngmid2 29248

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