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Theorem lmireu 26749

Description: Any point has a unique antecedent through line mirroring. Theorem 10.6 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
ismid.p	⊢ 𝑃 = (Base‘𝐺)
ismid.d	⊢ − = (dist‘𝐺)
ismid.i	⊢ 𝐼 = (Itv‘𝐺)
ismid.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
ismid.1	⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
lmif.m	⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
lmif.l	⊢ 𝐿 = (LineG‘𝐺)
lmif.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
lmicl.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)

**Assertion**
Ref	Expression
lmireu	⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 (𝑀‘𝑏) = 𝐴)

Distinct variable groups: 𝐺,𝑏 𝑃,𝑏 𝜑,𝑏 𝐷,𝑏 𝐿,𝑏 𝐴,𝑏 𝑀,𝑏 Allowed substitution hints: 𝐼(𝑏) − (𝑏)

**Proof of Theorem lmireu**
Step	Hyp	Ref	Expression
1		ismid.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		ismid.d	. . 3 ⊢ − = (dist‘𝐺)
3		ismid.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		ismid.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		ismid.1	. . 3 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
6		lmif.m	. . 3 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
7		lmif.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
8		lmif.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
9		lmicl.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	lmicl 26745	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
11	1, 2, 3, 4, 5, 6, 7, 8, 9	lmilmi 26748	. 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴)
12	4	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝐺 ∈ TarskiG)
13	5	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝐺Dim_TarskiG≥2)
14	8	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝐷 ∈ ran 𝐿)
15		simplr 769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝑏 ∈ 𝑃)
16	1, 2, 3, 12, 13, 6, 7, 14, 15	lmilmi 26748	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → (𝑀‘(𝑀‘𝑏)) = 𝑏)
17		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → (𝑀‘𝑏) = 𝐴)
18	17	fveq2d 6691	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → (𝑀‘(𝑀‘𝑏)) = (𝑀‘𝐴))
19	16, 18	eqtr3d 2776	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝑏 = (𝑀‘𝐴))
20	19	ex 416	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → ((𝑀‘𝑏) = 𝐴 → 𝑏 = (𝑀‘𝐴)))
21	20	ralrimiva 3097	. 2 ⊢ (𝜑 → ∀𝑏 ∈ 𝑃 ((𝑀‘𝑏) = 𝐴 → 𝑏 = (𝑀‘𝐴)))
22		fveqeq2 6696	. . 3 ⊢ (𝑏 = (𝑀‘𝐴) → ((𝑀‘𝑏) = 𝐴 ↔ (𝑀‘(𝑀‘𝐴)) = 𝐴))
23	22	eqreu 3633	. 2 ⊢ (((𝑀‘𝐴) ∈ 𝑃 ∧ (𝑀‘(𝑀‘𝐴)) = 𝐴 ∧ ∀𝑏 ∈ 𝑃 ((𝑀‘𝑏) = 𝐴 → 𝑏 = (𝑀‘𝐴))) → ∃!𝑏 ∈ 𝑃 (𝑀‘𝑏) = 𝐴)
24	10, 11, 21, 23	syl3anc 1372	1 ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 (𝑀‘𝑏) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ∃!wreu 3056 class class class wbr 5040 ran crn 5536 ‘cfv 6350 2c2 11784 Basecbs 16599 distcds 16690 TarskiGcstrkg 26389 Dim_TarskiG≥cstrkgld 26393 Itvcitv 26395 LineGclng 26396 lInvGclmi 26732

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 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-oadd 8148 df-er 8333 df-map 8452 df-pm 8453 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-dju 9416 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-3 11793 df-n0 11990 df-xnn0 12062 df-z 12076 df-uz 12338 df-fz 12995 df-fzo 13138 df-hash 13796 df-word 13969 df-concat 14025 df-s1 14052 df-s2 14312 df-s3 14313 df-trkgc 26407 df-trkgb 26408 df-trkgcb 26409 df-trkgld 26411 df-trkg 26412 df-cgrg 26470 df-leg 26542 df-mir 26612 df-rag 26653 df-perpg 26655 df-mid 26733 df-lmi 26734

This theorem is referenced by: lmieq 26750

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