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Theorem lmireu 28875

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 28871	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
11	1, 2, 3, 4, 5, 6, 7, 8, 9	lmilmi 28874	. 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴)
12	4	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝐺 ∈ TarskiG)
13	5	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝐺Dim_TarskiG≥2)
14	8	ad2antrr 727	. . . . . 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 28874	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → (𝑀‘(𝑀‘𝑏)) = 𝑏)
17		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → (𝑀‘𝑏) = 𝐴)
18	17	fveq2d 6839	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → (𝑀‘(𝑀‘𝑏)) = (𝑀‘𝐴))
19	16, 18	eqtr3d 2774	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝑏 = (𝑀‘𝐴))
20	19	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → ((𝑀‘𝑏) = 𝐴 → 𝑏 = (𝑀‘𝐴)))
21	20	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑏 ∈ 𝑃 ((𝑀‘𝑏) = 𝐴 → 𝑏 = (𝑀‘𝐴)))
22		fveqeq2 6844	. . 3 ⊢ (𝑏 = (𝑀‘𝐴) → ((𝑀‘𝑏) = 𝐴 ↔ (𝑀‘(𝑀‘𝐴)) = 𝐴))
23	22	eqreu 3676	. 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 1374	1 ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 (𝑀‘𝑏) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃!wreu 3341 class class class wbr 5086 ran crn 5626 ‘cfv 6493 2c2 12230 Basecbs 17173 distcds 17223 TarskiGcstrkg 28512 Dim_TarskiG≥cstrkgld 28516 Itvcitv 28518 LineGclng 28519 lInvGclmi 28858

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-dju 9819 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-n0 12432 df-xnn0 12505 df-z 12519 df-uz 12783 df-fz 13456 df-fzo 13603 df-hash 14287 df-word 14470 df-concat 14527 df-s1 14553 df-s2 14804 df-s3 14805 df-trkgc 28533 df-trkgb 28534 df-trkgcb 28535 df-trkgld 28537 df-trkg 28538 df-cgrg 28596 df-leg 28668 df-mir 28738 df-rag 28779 df-perpg 28781 df-mid 28859 df-lmi 28860

This theorem is referenced by: lmieq 28876

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