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Theorem lmiopp 29090

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 29073	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
12		eqidd 2764	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘𝐴))
13	1, 2, 3, 6, 7, 8, 9, 10, 5, 11	islmib 29074	. . . 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 235	. . 3 ⊢ (𝜑 → ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴))))
15	14	simpld 499	. 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 29076	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴)
18	17	eqeq1d 2765	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ 𝐴 = (𝑀‘𝐴)))
19	1, 2, 3, 6, 7, 8, 9, 10, 11	lmiinv 29079	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ (𝑀‘𝐴) ∈ 𝐷))
20		eqcom 2770	. . . . . 6 ⊢ (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴)
21	20	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴))
22	18, 19, 21	3bitr3d 312	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ (𝑀‘𝐴) = 𝐴))
23	1, 2, 3, 6, 7, 8, 9, 10, 5	lmiinv 29079	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷))
24	22, 23	bitrd 282	. . 3 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ 𝐴 ∈ 𝐷))
25	16, 24	mtbird 328	. 2 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ 𝐷)
26	1, 2, 3, 6, 7, 5, 11	midbtwn 29066	. 2 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴)))
27	1, 2, 3, 4, 5, 11, 15, 16, 25, 26	islnoppd 28999	1 ⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ∃wrex 3089 ∖ cdif 3903 class class class wbr 5110 {copab 5174 ran crn 5664 ‘cfv 6538 (class class class)co 7412 2c2 12296 Basecbs 17270 distcds 17320 TarskiGcstrkg 28674 Dim_TarskiG≥cstrkgld 28678 Itvcitv 28680 LineGclng 28681 ⟂Gcperpg 28953 midGcmid 29059 lInvGclmi 29060

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-oadd 8458 df-er 8695 df-map 8827 df-pm 8828 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-dju 9888 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-xnn0 12579 df-z 12593 df-uz 12864 df-fz 13537 df-fzo 13685 df-hash 14369 df-word 14553 df-concat 14610 df-s1 14636 df-s2 14887 df-s3 14888 df-trkgc 28695 df-trkgb 28696 df-trkgcb 28697 df-trkgld 28699 df-trkg 28700 df-cgrg 28758 df-leg 28830 df-mir 28908 df-rag 28952 df-perpg 28954 df-mid 29061 df-lmi 29062

This theorem is referenced by: trgcopyeulem 29094

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