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Theorem lmimid 29300

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 6887	. . . . 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 2760	. . . . . . 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 29013	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
15	4, 5, 6, 10, 11, 7, 14, 1, 9	mircl 29134	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐶) ∈ 𝑃)
16	4, 5, 6, 7, 8, 9, 15, 11, 14	ismidb 29284	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐶) = (((pInvG‘𝐺)‘𝐵)‘𝐶) ↔ (𝐶(midG‘𝐺)(𝑆‘𝐶)) = 𝐵))
17	3, 16	mpbid 235	. . . 4 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) = 𝐵)
18	17, 13	eqeltrd 2860	. . 3 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) ∈ 𝐷)
19		df-ne 2956	. . . . . 6 ⊢ (𝐶 ≠ (𝑆‘𝐶) ↔ ¬ 𝐶 = (𝑆‘𝐶))
20	7	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐺 ∈ TarskiG)
21	12	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐷 ∈ ran 𝐿)
22	9	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐶 ∈ 𝑃)
23	15	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → (𝑆‘𝐶) ∈ 𝑃)
24		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐶 ≠ (𝑆‘𝐶))
25	4, 6, 10, 20, 22, 23, 24	tgelrnln 29099	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → (𝐶𝐿(𝑆‘𝐶)) ∈ ran 𝐿)
26	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ 𝐷)
27	14	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ 𝑃)
28	4, 5, 6, 7, 8, 9, 15	midbtwn 29285	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) ∈ (𝐶𝐼(𝑆‘𝐶)))
29	17, 28	eqeltrrd 2861	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼(𝑆‘𝐶)))
30	29	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ (𝐶𝐼(𝑆‘𝐶)))
31	4, 6, 10, 20, 22, 23, 27, 24, 30	btwnlng1 29088	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ (𝐶𝐿(𝑆‘𝐶)))
32	26, 31	elind 4097	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ (𝐷 ∩ (𝐶𝐿(𝑆‘𝐶))))
33		lmimid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
34	33	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐴 ∈ 𝐷)
35	4, 6, 10, 20, 22, 23, 24	tglinerflx1 29102	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐶 ∈ (𝐶𝐿(𝑆‘𝐶)))
36		lmimid.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
37	36	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐴 ≠ 𝐵)
38	4, 5, 6, 10, 11, 7, 14, 1, 9	mirinv 29139	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
39		eqcom 2767	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
40	38, 39	bitrdi 290	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆‘𝐶) = 𝐶 ↔ 𝐶 = 𝐵))
41	40	biimpar 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 = 𝐵) → (𝑆‘𝐶) = 𝐶)
42	41	eqcomd 2766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐵) → 𝐶 = (𝑆‘𝐶))
43	42	ex 418	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 = 𝐵 → 𝐶 = (𝑆‘𝐶)))
44	43	necon3d 2976	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ≠ (𝑆‘𝐶) → 𝐶 ≠ 𝐵))
45	44	imp 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐶 ≠ 𝐵)
46		lmimid.r	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
47	46	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
48	4, 5, 6, 10, 20, 21, 25, 32, 34, 35, 37, 45, 47	ragperp 29193	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶)))
49	48	ex 418	. . . . . 6 ⊢ (𝜑 → (𝐶 ≠ (𝑆‘𝐶) → 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶))))
50	19, 49	biimtrrid 246	. . . . 5 ⊢ (𝜑 → (¬ 𝐶 = (𝑆‘𝐶) → 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶))))
51	50	orrd 877	. . . 4 ⊢ (𝜑 → (𝐶 = (𝑆‘𝐶) ∨ 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶))))
52	51	orcomd 885	. . 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 29293	. . 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 726	. 2 ⊢ (𝜑 → (𝑆‘𝐶) = (𝑀‘𝐶))
56	55	eqcomd 2766	1 ⊢ (𝜑 → (𝑀‘𝐶) = (𝑆‘𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ran crn 5651 ‘cfv 6538 (class class class)co 7420 2c2 12398 ⟨“cs3 14994 Basecbs 17388 distcds 17438 TarskiGcstrkg 28890 Dim_TarskiG≥cstrkgld 28894 Itvcitv 28896 LineGclng 28897 pInvGcmir 29125 ∟Gcrag 29169 ⟂Gcperpg 29171 midGcmid 29278 lInvGclmi 29279

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 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 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-oadd 8481 df-er 8718 df-map 8850 df-pm 8851 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-dju 9983 df-card 10021 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-nn 12337 df-2 12406 df-3 12407 df-n0 12608 df-xnn0 12681 df-z 12695 df-uz 12967 df-fz 13641 df-fzo 13790 df-hash 14476 df-word 14660 df-concat 14717 df-s1 14744 df-s2 15000 df-s3 15001 df-trkgc 28911 df-trkgb 28912 df-trkgcb 28913 df-trkgld 28915 df-trkg 28916 df-cgrg 28975 df-leg 29047 df-mir 29126 df-rag 29170 df-perpg 29172 df-mid 29280 df-lmi 29281

This theorem is used by: hypcgrlem1 29306

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