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Theorem lmireu 29304

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 29300	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
11	1, 2, 3, 4, 5, 6, 7, 8, 9	lmilmi 29303	. 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴)
12	4	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝐺 ∈ TarskiG)
13	5	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝐺Dim_TarskiG≥2)
14	8	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝐷 ∈ ran 𝐿)
15		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝑏 ∈ 𝑃)
16	1, 2, 3, 12, 13, 6, 7, 14, 15	lmilmi 29303	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → (𝑀‘(𝑀‘𝑏)) = 𝑏)
17		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → (𝑀‘𝑏) = 𝐴)
18	17	fveq2d 6894	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → (𝑀‘(𝑀‘𝑏)) = (𝑀‘𝐴))
19	16, 18	eqtr3d 2797	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝑏 = (𝑀‘𝐴))
20	19	ex 418	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → ((𝑀‘𝑏) = 𝐴 → 𝑏 = (𝑀‘𝐴)))
21	20	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑏 ∈ 𝑃 ((𝑀‘𝑏) = 𝐴 → 𝑏 = (𝑀‘𝐴)))
22		fveqeq2 6899	. . 3 ⊢ (𝑏 = (𝑀‘𝐴) → ((𝑀‘𝑏) = 𝐴 ↔ (𝑀‘(𝑀‘𝐴)) = 𝐴))
23	22	eqreu 3686	. 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 1398	1 ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 (𝑀‘𝑏) = 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ∃!wreu 3363 class class class wbr 5102 ran crn 5652 ‘cfv 6543 2c2 12404 Basecbs 17396 distcds 17446 TarskiGcstrkg 28898 Dim_TarskiG≥cstrkgld 28902 Itvcitv 28904 LineGclng 28905 lInvGclmi 29287

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-oadd 8488 df-er 8725 df-map 8857 df-pm 8858 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-dju 9989 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-nn 12343 df-2 12412 df-3 12413 df-n0 12614 df-xnn0 12687 df-z 12701 df-uz 12973 df-fz 13647 df-fzo 13796 df-hash 14482 df-word 14666 df-concat 14723 df-s1 14750 df-s2 15007 df-s3 15008 df-trkgc 28919 df-trkgb 28920 df-trkgcb 28921 df-trkgld 28923 df-trkg 28924 df-cgrg 28983 df-leg 29055 df-mir 29134 df-rag 29178 df-perpg 29180 df-mid 29288 df-lmi 29289

This theorem is used by: lmieq 29305

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