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Theorem lmiopp 28778

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 28762	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
12		eqidd 2732	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘𝐴))
13	1, 2, 3, 6, 7, 8, 9, 10, 5, 11	islmib 28763	. . . 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 232	. . 3 ⊢ (𝜑 → ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴))))
15	14	simpld 494	. 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 28765	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴)
18	17	eqeq1d 2733	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ 𝐴 = (𝑀‘𝐴)))
19	1, 2, 3, 6, 7, 8, 9, 10, 11	lmiinv 28768	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ (𝑀‘𝐴) ∈ 𝐷))
20		eqcom 2738	. . . . . 6 ⊢ (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴)
21	20	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴))
22	18, 19, 21	3bitr3d 309	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ (𝑀‘𝐴) = 𝐴))
23	1, 2, 3, 6, 7, 8, 9, 10, 5	lmiinv 28768	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷))
24	22, 23	bitrd 279	. . 3 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ 𝐴 ∈ 𝐷))
25	16, 24	mtbird 325	. 2 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ 𝐷)
26	1, 2, 3, 6, 7, 5, 11	midbtwn 28755	. 2 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴)))
27	1, 2, 3, 4, 5, 11, 15, 16, 25, 26	islnoppd 28716	1 ⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2111 ∃wrex 3056 ∖ cdif 3899 class class class wbr 5091 {copab 5153 ran crn 5617 ‘cfv 6481 (class class class)co 7346 2c2 12177 Basecbs 17117 distcds 17167 TarskiGcstrkg 28403 Dim_TarskiG≥cstrkgld 28407 Itvcitv 28409 LineGclng 28410 ⟂Gcperpg 28671 midGcmid 28748 lInvGclmi 28749

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-oadd 8389 df-er 8622 df-map 8752 df-pm 8753 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-dju 9791 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-nn 12123 df-2 12185 df-3 12186 df-n0 12379 df-xnn0 12452 df-z 12466 df-uz 12730 df-fz 13405 df-fzo 13552 df-hash 14235 df-word 14418 df-concat 14475 df-s1 14501 df-s2 14752 df-s3 14753 df-trkgc 28424 df-trkgb 28425 df-trkgcb 28426 df-trkgld 28428 df-trkg 28429 df-cgrg 28487 df-leg 28559 df-mir 28629 df-rag 28670 df-perpg 28672 df-mid 28750 df-lmi 28751

This theorem is referenced by: trgcopyeulem 28781

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