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Theorem lmimid 28728

Description: If we have a right angle, then the mirror point is the point inversion. (Contributed by Thierry Arnoux, 15-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)
lmimid.s	⊢ 𝑆 = ((pInvG‘𝐺)‘𝐵)
lmimid.r	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
lmimid.a	⊢ (𝜑 → 𝐴 ∈ 𝐷)
lmimid.b	⊢ (𝜑 → 𝐵 ∈ 𝐷)
lmimid.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
lmimid.d	⊢ (𝜑 → 𝐴 ≠ 𝐵)

**Assertion**
Ref	Expression
lmimid	⊢ (𝜑 → (𝑀‘𝐶) = (𝑆‘𝐶))

**Proof of Theorem lmimid**
Step	Hyp	Ref	Expression
1		lmimid.s	. . . . . . 7 ⊢ 𝑆 = ((pInvG‘𝐺)‘𝐵)
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑆 = ((pInvG‘𝐺)‘𝐵))
3	2	fveq1d 6863	. . . . 5 ⊢ (𝜑 → (𝑆‘𝐶) = (((pInvG‘𝐺)‘𝐵)‘𝐶))
4		ismid.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
5		ismid.d	. . . . . 6 ⊢ − = (dist‘𝐺)
6		ismid.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
7		ismid.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8		ismid.1	. . . . . 6 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
9		lmimid.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
10		lmif.l	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
11		eqid 2730	. . . . . . 7 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
12		lmif.d	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
13		lmimid.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
14	4, 10, 6, 7, 12, 13	tglnpt 28483	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
15	4, 5, 6, 10, 11, 7, 14, 1, 9	mircl 28595	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐶) ∈ 𝑃)
16	4, 5, 6, 7, 8, 9, 15, 11, 14	ismidb 28712	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐶) = (((pInvG‘𝐺)‘𝐵)‘𝐶) ↔ (𝐶(midG‘𝐺)(𝑆‘𝐶)) = 𝐵))
17	3, 16	mpbid 232	. . . 4 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) = 𝐵)
18	17, 13	eqeltrd 2829	. . 3 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) ∈ 𝐷)
19		df-ne 2927	. . . . . 6 ⊢ (𝐶 ≠ (𝑆‘𝐶) ↔ ¬ 𝐶 = (𝑆‘𝐶))
20	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐺 ∈ TarskiG)
21	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐷 ∈ ran 𝐿)
22	9	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐶 ∈ 𝑃)
23	15	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → (𝑆‘𝐶) ∈ 𝑃)
24		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐶 ≠ (𝑆‘𝐶))
25	4, 6, 10, 20, 22, 23, 24	tgelrnln 28564	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → (𝐶𝐿(𝑆‘𝐶)) ∈ ran 𝐿)
26	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ 𝐷)
27	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ 𝑃)
28	4, 5, 6, 7, 8, 9, 15	midbtwn 28713	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) ∈ (𝐶𝐼(𝑆‘𝐶)))
29	17, 28	eqeltrrd 2830	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼(𝑆‘𝐶)))
30	29	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ (𝐶𝐼(𝑆‘𝐶)))
31	4, 6, 10, 20, 22, 23, 27, 24, 30	btwnlng1 28553	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ (𝐶𝐿(𝑆‘𝐶)))
32	26, 31	elind 4166	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ (𝐷 ∩ (𝐶𝐿(𝑆‘𝐶))))
33		lmimid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
34	33	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐴 ∈ 𝐷)
35	4, 6, 10, 20, 22, 23, 24	tglinerflx1 28567	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐶 ∈ (𝐶𝐿(𝑆‘𝐶)))
36		lmimid.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
37	36	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐴 ≠ 𝐵)
38	4, 5, 6, 10, 11, 7, 14, 1, 9	mirinv 28600	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
39		eqcom 2737	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
40	38, 39	bitrdi 287	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆‘𝐶) = 𝐶 ↔ 𝐶 = 𝐵))
41	40	biimpar 477	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 = 𝐵) → (𝑆‘𝐶) = 𝐶)
42	41	eqcomd 2736	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐵) → 𝐶 = (𝑆‘𝐶))
43	42	ex 412	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 = 𝐵 → 𝐶 = (𝑆‘𝐶)))
44	43	necon3d 2947	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ≠ (𝑆‘𝐶) → 𝐶 ≠ 𝐵))
45	44	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐶 ≠ 𝐵)
46		lmimid.r	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
47	46	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
48	4, 5, 6, 10, 20, 21, 25, 32, 34, 35, 37, 45, 47	ragperp 28651	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶)))
49	48	ex 412	. . . . . 6 ⊢ (𝜑 → (𝐶 ≠ (𝑆‘𝐶) → 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶))))
50	19, 49	biimtrrid 243	. . . . 5 ⊢ (𝜑 → (¬ 𝐶 = (𝑆‘𝐶) → 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶))))
51	50	orrd 863	. . . 4 ⊢ (𝜑 → (𝐶 = (𝑆‘𝐶) ∨ 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶))))
52	51	orcomd 871	. . 3 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶)) ∨ 𝐶 = (𝑆‘𝐶)))
53		lmif.m	. . . 4 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
54	4, 5, 6, 7, 8, 53, 10, 12, 9, 15	islmib 28721	. . 3 ⊢ (𝜑 → ((𝑆‘𝐶) = (𝑀‘𝐶) ↔ ((𝐶(midG‘𝐺)(𝑆‘𝐶)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶)) ∨ 𝐶 = (𝑆‘𝐶)))))
55	18, 52, 54	mpbir2and 713	. 2 ⊢ (𝜑 → (𝑆‘𝐶) = (𝑀‘𝐶))
56	55	eqcomd 2736	1 ⊢ (𝜑 → (𝑀‘𝐶) = (𝑆‘𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 class class class wbr 5110 ran crn 5642 ‘cfv 6514 (class class class)co 7390 2c2 12248 ⟨“cs3 14815 Basecbs 17186 distcds 17236 TarskiGcstrkg 28361 Dim_TarskiG≥cstrkgld 28365 Itvcitv 28367 LineGclng 28368 pInvGcmir 28586 ∟Gcrag 28627 ⟂Gcperpg 28629 midGcmid 28706 lInvGclmi 28707

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-oadd 8441 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-xnn0 12523 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-hash 14303 df-word 14486 df-concat 14543 df-s1 14568 df-s2 14821 df-s3 14822 df-trkgc 28382 df-trkgb 28383 df-trkgcb 28384 df-trkgld 28386 df-trkg 28387 df-cgrg 28445 df-leg 28517 df-mir 28587 df-rag 28628 df-perpg 28630 df-mid 28708 df-lmi 28709

This theorem is referenced by: hypcgrlem1 28733

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