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Theorem lmiopp 26167

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.1	. . . 4 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
2		lmiopp.p	. . . . . . . . 9 ⊢ 𝑃 = (Base‘𝐺)
3		lmiopp.m	. . . . . . . . 9 ⊢ − = (dist‘𝐺)
4		lmiopp.i	. . . . . . . . 9 ⊢ 𝐼 = (Itv‘𝐺)
5		lmiopp.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		lmiopp.h	. . . . . . . . 9 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
7		lmiopp.n	. . . . . . . . 9 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
8		lmiopp.l	. . . . . . . . 9 ⊢ 𝐿 = (LineG‘𝐺)
9		lmiopp.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
10		lmiopp.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11	2, 3, 4, 5, 6, 7, 8, 9, 10	lmilmi 26154	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴)
12	11	eqeq1d 2780	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ 𝐴 = (𝑀‘𝐴)))
13	2, 3, 4, 5, 6, 7, 8, 9, 10	lmicl 26151	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
14	2, 3, 4, 5, 6, 7, 8, 9, 13	lmiinv 26157	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ (𝑀‘𝐴) ∈ 𝐷))
15		eqcom 2785	. . . . . . . 8 ⊢ (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴)
16	15	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴))
17	12, 14, 16	3bitr3d 301	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ (𝑀‘𝐴) = 𝐴))
18	2, 3, 4, 5, 6, 7, 8, 9, 10	lmiinv 26157	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷))
19	17, 18	bitrd 271	. . . . 5 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ 𝐴 ∈ 𝐷))
20	1, 19	mtbird 317	. . . 4 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ 𝐷)
21	1, 20	jca 507	. . 3 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐷 ∧ ¬ (𝑀‘𝐴) ∈ 𝐷))
22		eqidd 2779	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘𝐴))
23	2, 3, 4, 5, 6, 7, 8, 9, 10, 13	islmib 26152	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐴) = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴)))))
24	22, 23	mpbid 224	. . . . 5 ⊢ (𝜑 → ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴))))
25	24	simpld 490	. . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷)
26	2, 3, 4, 5, 6, 10, 13	midbtwn 26144	. . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴)))
27		eleq1 2847	. . . . 5 ⊢ (𝑡 = (𝐴(midG‘𝐺)(𝑀‘𝐴)) → (𝑡 ∈ (𝐴𝐼(𝑀‘𝐴)) ↔ (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴))))
28	27	rspcev 3511	. . . 4 ⊢ (((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴))) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼(𝑀‘𝐴)))
29	25, 26, 28	syl2anc 579	. . 3 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼(𝑀‘𝐴)))
30	21, 29	jca 507	. 2 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ (𝑀‘𝐴) ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼(𝑀‘𝐴))))
31		lmiopp.o	. . 3 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
32	2, 3, 4, 31, 10, 13	islnopp 26104	. 2 ⊢ (𝜑 → (𝐴𝑂(𝑀‘𝐴) ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ (𝑀‘𝐴) ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼(𝑀‘𝐴)))))
33	30, 32	mpbird 249	1 ⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 ∖ cdif 3789 class class class wbr 4888 {copab 4950 ran crn 5358 ‘cfv 6137 (class class class)co 6924 2c2 11435 Basecbs 16266 distcds 16358 TarskiGcstrkg 25798 Dim_TarskiG≥cstrkgld 25802 Itvcitv 25804 LineGclng 25805 ⟂Gcperpg 26063 midGcmid 26137 lInvGclmi 26138

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-xnn0 11720 df-z 11734 df-uz 11998 df-fz 12649 df-fzo 12790 df-hash 13442 df-word 13606 df-concat 13667 df-s1 13692 df-s2 14005 df-s3 14006 df-trkgc 25816 df-trkgb 25817 df-trkgcb 25818 df-trkgld 25820 df-trkg 25821 df-cgrg 25879 df-leg 25951 df-mir 26021 df-rag 26062 df-perpg 26064 df-mid 26139 df-lmi 26140

This theorem is referenced by: trgcopyeulem 26170

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