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Theorem lmimid 28887

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 6836	. . . . 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 2740	. . . . . . 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 28642	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
15	4, 5, 6, 10, 11, 7, 14, 1, 9	mircl 28754	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐶) ∈ 𝑃)
16	4, 5, 6, 7, 8, 9, 15, 11, 14	ismidb 28871	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐶) = (((pInvG‘𝐺)‘𝐵)‘𝐶) ↔ (𝐶(midG‘𝐺)(𝑆‘𝐶)) = 𝐵))
17	3, 16	mpbid 233	. . . 4 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) = 𝐵)
18	17, 13	eqeltrd 2840	. . 3 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) ∈ 𝐷)
19		df-ne 2936	. . . . . 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 28723	. . . . . . . 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 28872	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) ∈ (𝐶𝐼(𝑆‘𝐶)))
29	17, 28	eqeltrrd 2841	. . . . . . . . . . 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 28712	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ (𝐶𝐿(𝑆‘𝐶)))
32	26, 31	elind 4136	. . . . . . . 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 28726	. . . . . . . 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 28759	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
39		eqcom 2747	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝐶 ↔ 𝐶 = 𝐵)
40	38, 39	bitrdi 288	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆‘𝐶) = 𝐶 ↔ 𝐶 = 𝐵))
41	40	biimpar 478	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 = 𝐵) → (𝑆‘𝐶) = 𝐶)
42	41	eqcomd 2746	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐵) → 𝐶 = (𝑆‘𝐶))
43	42	ex 413	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 = 𝐵 → 𝐶 = (𝑆‘𝐶)))
44	43	necon3d 2956	. . . . . . . . 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 28810	. . . . . . 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 244	. . . . 5 ⊢ (𝜑 → (¬ 𝐶 = (𝑆‘𝐶) → 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶))))
51	50	orrd 869	. . . 4 ⊢ (𝜑 → (𝐶 = (𝑆‘𝐶) ∨ 𝐷(⟂G‘𝐺)(𝐶𝐿(𝑆‘𝐶))))
52	51	orcomd 877	. . 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 28880	. . 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 719	. 2 ⊢ (𝜑 → (𝑆‘𝐶) = (𝑀‘𝐶))
56	55	eqcomd 2746	1 ⊢ (𝜑 → (𝑀‘𝐶) = (𝑆‘𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 class class class wbr 5079 ran crn 5626 ‘cfv 6492 (class class class)co 7363 2c2 12234 ⟨“cs3 14802 Basecbs 17177 distcds 17227 TarskiGcstrkg 28520 Dim_TarskiG≥cstrkgld 28524 Itvcitv 28526 LineGclng 28527 pInvGcmir 28745 ∟Gcrag 28786 ⟂Gcperpg 28788 midGcmid 28865 lInvGclmi 28866

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-oadd 8406 df-er 8640 df-map 8772 df-pm 8773 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9823 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-n0 12436 df-xnn0 12509 df-z 12523 df-uz 12787 df-fz 13460 df-fzo 13607 df-hash 14291 df-word 14474 df-concat 14531 df-s1 14557 df-s2 14808 df-s3 14809 df-trkgc 28541 df-trkgb 28542 df-trkgcb 28543 df-trkgld 28545 df-trkg 28546 df-cgrg 28604 df-leg 28676 df-mir 28746 df-rag 28787 df-perpg 28789 df-mid 28867 df-lmi 28868

This theorem is referenced by: hypcgrlem1 28892

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