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Theorem lmimid 28042

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 6893	. . . . 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 2732	. . . . . . 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 27797	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
15	4, 5, 6, 10, 11, 7, 14, 1, 9	mircl 27909	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐶) ∈ 𝑃)
16	4, 5, 6, 7, 8, 9, 15, 11, 14	ismidb 28026	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐶) = (((pInvG‘𝐺)‘𝐵)‘𝐶) ↔ (𝐶(midG‘𝐺)(𝑆‘𝐶)) = 𝐵))
17	3, 16	mpbid 231	. . . 4 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) = 𝐵)
18	17, 13	eqeltrd 2833	. . 3 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) ∈ 𝐷)
19		df-ne 2941	. . . . . 6 ⊢ (𝐶 ≠ (𝑆‘𝐶) ↔ ¬ 𝐶 = (𝑆‘𝐶))
20	7	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐺 ∈ TarskiG)
21	12	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐷 ∈ ran 𝐿)
22	9	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐶 ∈ 𝑃)
23	15	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → (𝑆‘𝐶) ∈ 𝑃)
24		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐶 ≠ (𝑆‘𝐶))
25	4, 6, 10, 20, 22, 23, 24	tgelrnln 27878	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → (𝐶𝐿(𝑆‘𝐶)) ∈ ran 𝐿)
26	13	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ 𝐷)
27	14	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ 𝑃)
28	4, 5, 6, 7, 8, 9, 15	midbtwn 28027	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) ∈ (𝐶𝐼(𝑆‘𝐶)))
29	17, 28	eqeltrrd 2834	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼(𝑆‘𝐶)))
30	29	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ (𝐶𝐼(𝑆‘𝐶)))
31	4, 6, 10, 20, 22, 23, 27, 24, 30	btwnlng1 27867	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ (𝐶𝐿(𝑆‘𝐶)))
32	26, 31	elind 4194	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ (𝐷 ∩ (𝐶𝐿(𝑆‘𝐶))))
33		lmimid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
34	33	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐴 ∈ 𝐷)
35	4, 6, 10, 20, 22, 23, 24	tglinerflx1 27881	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐶 ∈ (𝐶𝐿(𝑆‘𝐶)))
36		lmimid.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
37	36	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐴 ≠ 𝐵)
38	4, 5, 6, 10, 11, 7, 14, 1, 9	mirinv 27914	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
39		eqcom 2739	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
40	38, 39	bitrdi 286	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆‘𝐶) = 𝐶 ↔ 𝐶 = 𝐵))
41	40	biimpar 478	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 = 𝐵) → (𝑆‘𝐶) = 𝐶)
42	41	eqcomd 2738	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐵) → 𝐶 = (𝑆‘𝐶))
43	42	ex 413	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 = 𝐵 → 𝐶 = (𝑆‘𝐶)))
44	43	necon3d 2961	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ≠ (𝑆‘𝐶) → 𝐶 ≠ 𝐵))
45	44	imp 407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐶 ≠ 𝐵)
46		lmimid.r	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
47	46	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
48	4, 5, 6, 10, 20, 21, 25, 32, 34, 35, 37, 45, 47	ragperp 27965	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶)))
49	48	ex 413	. . . . . 6 ⊢ (𝜑 → (𝐶 ≠ (𝑆‘𝐶) → 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶))))
50	19, 49	biimtrrid 242	. . . . 5 ⊢ (𝜑 → (¬ 𝐶 = (𝑆‘𝐶) → 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶))))
51	50	orrd 861	. . . 4 ⊢ (𝜑 → (𝐶 = (𝑆‘𝐶) ∨ 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶))))
52	51	orcomd 869	. . 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 28035	. . 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 711	. 2 ⊢ (𝜑 → (𝑆‘𝐶) = (𝑀‘𝐶))
56	55	eqcomd 2738	1 ⊢ (𝜑 → (𝑀‘𝐶) = (𝑆‘𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5148 ran crn 5677 ‘cfv 6543 (class class class)co 7408 2c2 12266 ⟨“cs3 14792 Basecbs 17143 distcds 17205 TarskiGcstrkg 27675 Dim_TarskiG≥cstrkgld 27679 Itvcitv 27681 LineGclng 27682 pInvGcmir 27900 ∟Gcrag 27941 ⟂Gcperpg 27943 midGcmid 28020 lInvGclmi 28021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-dju 9895 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-xnn0 12544 df-z 12558 df-uz 12822 df-fz 13484 df-fzo 13627 df-hash 14290 df-word 14464 df-concat 14520 df-s1 14545 df-s2 14798 df-s3 14799 df-trkgc 27696 df-trkgb 27697 df-trkgcb 27698 df-trkgld 27700 df-trkg 27701 df-cgrg 27759 df-leg 27831 df-mir 27901 df-rag 27942 df-perpg 27944 df-mid 28022 df-lmi 28023

This theorem is referenced by: hypcgrlem1 28047

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