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Theorem lmimid 29179

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 6884	. . . . 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 2762	. . . . . . 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 28892	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
15	4, 5, 6, 10, 11, 7, 14, 1, 9	mircl 29013	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐶) ∈ 𝑃)
16	4, 5, 6, 7, 8, 9, 15, 11, 14	ismidb 29163	. . . . 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 2862	. . 3 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) ∈ 𝐷)
19		df-ne 2958	. . . . . 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 28978	. . . . . . . 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 29164	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶(midG‘𝐺)(𝑆‘𝐶)) ∈ (𝐶𝐼(𝑆‘𝐶)))
29	17, 28	eqeltrrd 2863	. . . . . . . . . . 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 28967	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 ≠ (𝑆‘𝐶)) → 𝐵 ∈ (𝐶𝐿(𝑆‘𝐶)))
32	26, 31	elind 4149	. . . . . . . 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 28981	. . . . . . . 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 29018	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶))
39		eqcom 2769	. . . . . . . . . . . . . 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 2768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐵) → 𝐶 = (𝑆‘𝐶))
43	42	ex 418	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 = 𝐵 → 𝐶 = (𝑆‘𝐶)))
44	43	necon3d 2978	. . . . . . . . 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 29072	. . . . . . 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 29172	. . 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 2768	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 1570 ∈ wcel 2145 ≠ wne 2957 class class class wbr 5107 ran crn 5660 ‘cfv 6537 (class class class)co 7416 2c2 12322 ⟨“cs3 14915 Basecbs 17305 distcds 17355 TarskiGcstrkg 28769 Dim_TarskiG≥cstrkgld 28773 Itvcitv 28775 LineGclng 28776 pInvGcmir 29004 ∟Gcrag 29048 ⟂Gcperpg 29050 midGcmid 29157 lInvGclmi 29158

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-oadd 8462 df-er 8699 df-map 8831 df-pm 8832 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9909 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-n0 12532 df-xnn0 12605 df-z 12619 df-uz 12891 df-fz 13564 df-fzo 13712 df-hash 14397 df-word 14581 df-concat 14638 df-s1 14665 df-s2 14921 df-s3 14922 df-trkgc 28790 df-trkgb 28791 df-trkgcb 28792 df-trkgld 28794 df-trkg 28795 df-cgrg 28854 df-leg 28926 df-mir 29005 df-rag 29049 df-perpg 29051 df-mid 29159 df-lmi 29160

This theorem is used by: hypcgrlem1 29185

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