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Theorem lmiopp 28042

Description: Line mirroring produces points on the opposite side of the mirroring line. Theorem 10.14 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 2-Aug-2020.)

**Hypotheses**
Ref	Expression
lmiopp.p	⊢ 𝑃 = (Base‘𝐺)
lmiopp.m	⊢ − = (dist‘𝐺)
lmiopp.i	⊢ 𝐼 = (Itv‘𝐺)
lmiopp.l	⊢ 𝐿 = (LineG‘𝐺)
lmiopp.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
lmiopp.h	⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
lmiopp.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
lmiopp.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
lmiopp.n	⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
lmiopp.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
lmiopp.1	⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)

**Assertion**
Ref	Expression
lmiopp	⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴))

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

**Proof of Theorem lmiopp**
Step	Hyp	Ref	Expression
1		lmiopp.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		lmiopp.m	. 2 ⊢ − = (dist‘𝐺)
3		lmiopp.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		lmiopp.o	. 2 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
5		lmiopp.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
6		lmiopp.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		lmiopp.h	. . 3 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
8		lmiopp.n	. . 3 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
9		lmiopp.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
10		lmiopp.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
11	1, 2, 3, 6, 7, 8, 9, 10, 5	lmicl 28026	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
12		eqidd 2733	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘𝐴))
13	1, 2, 3, 6, 7, 8, 9, 10, 5, 11	islmib 28027	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴)))))
14	12, 13	mpbid 231	. . 3 ⊢ (𝜑 → ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴))))
15	14	simpld 495	. 2 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷)
16		lmiopp.1	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
17	1, 2, 3, 6, 7, 8, 9, 10, 5	lmilmi 28029	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴)
18	17	eqeq1d 2734	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ 𝐴 = (𝑀‘𝐴)))
19	1, 2, 3, 6, 7, 8, 9, 10, 11	lmiinv 28032	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ (𝑀‘𝐴) ∈ 𝐷))
20		eqcom 2739	. . . . . 6 ⊢ (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴)
21	20	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴))
22	18, 19, 21	3bitr3d 308	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ (𝑀‘𝐴) = 𝐴))
23	1, 2, 3, 6, 7, 8, 9, 10, 5	lmiinv 28032	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷))
24	22, 23	bitrd 278	. . 3 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ 𝐴 ∈ 𝐷))
25	16, 24	mtbird 324	. 2 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ 𝐷)
26	1, 2, 3, 6, 7, 5, 11	midbtwn 28019	. 2 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴)))
27	1, 2, 3, 4, 5, 11, 15, 16, 25, 26	islnoppd 27980	1 ⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ∖ cdif 3944 class class class wbr 5147 {copab 5209 ran crn 5676 ‘cfv 6540 (class class class)co 7405 2c2 12263 Basecbs 17140 distcds 17202 TarskiGcstrkg 27667 Dim_TarskiG≥cstrkgld 27671 Itvcitv 27673 LineGclng 27674 ⟂Gcperpg 27935 midGcmid 28012 lInvGclmi 28013

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-s2 14795 df-s3 14796 df-trkgc 27688 df-trkgb 27689 df-trkgcb 27690 df-trkgld 27692 df-trkg 27693 df-cgrg 27751 df-leg 27823 df-mir 27893 df-rag 27934 df-perpg 27936 df-mid 28014 df-lmi 28015

This theorem is referenced by: trgcopyeulem 28045

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