Theorem nhpmirhp 29333

Description: If a point 𝑍 is on the plane defined by a line 𝐴 and a point 𝑌, but not on the same half-plane as 𝑌, then its mirror point (𝑀‘𝑍) by a point 𝑋 on 𝐴 is on the same half-plane as 𝑌. (Contributed by Thierry Arnoux, 5-Jul-2026.)

Hypotheses

Ref	Expression
nhpmirhp.p	⊢ 𝑃 = (Base‘𝐺)
nhpmirhp.l	⊢ 𝐿 = (LineG‘𝐺)
nhpmirhp.s	⊢ 𝑆 = (pInvG‘𝐺)
nhpmirhp.e	⊢ 𝐸 = (hlG‘𝐺)
nhpmirhp.m	⊢ 𝑀 = (𝑆‘𝑋)
nhpmirhp.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
nhpmirhp.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
nhpmirhp.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
nhpmirhp.y	⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ 𝐴))
nhpmirhp.z	⊢ (𝜑 → 𝑍 ∈ ((𝐴𝐸𝑌) ∖ 𝐴))
nhpmirhp.1	⊢ (𝜑 → ¬ 𝑌((hpG‘𝐺)‘𝐴)𝑍)

Assertion

Ref	Expression
nhpmirhp	⊢ (𝜑 → 𝑌((hpG‘𝐺)‘𝐴)(𝑀‘𝑍))

Proof of Theorem nhpmirhp

Dummy variables 𝑠 𝑡 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nhpmirhp.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2763	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
3		eqid 2763	. . 3 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
4		eleq1w 2847	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑃 ∖ 𝐴) ↔ 𝑧 ∈ (𝑃 ∖ 𝐴)))
5		eleq1w 2847	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝑃 ∖ 𝐴) ↔ 𝑤 ∈ (𝑃 ∖ 𝐴)))
6	4, 5	bi2anan9 650	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ↔ (𝑧 ∈ (𝑃 ∖ 𝐴) ∧ 𝑤 ∈ (𝑃 ∖ 𝐴))))
7		oveq12 7460	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥(Itv‘𝐺)𝑦) = (𝑧(Itv‘𝐺)𝑤))
8	7	eleq2d 2850	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑠 ∈ (𝑧(Itv‘𝐺)𝑤)))
9	8	rexbidv 3190	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑧(Itv‘𝐺)𝑤)))
10		eleq1w 2847	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ (𝑧(Itv‘𝐺)𝑤) ↔ 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤)))
11	10	cbvrexvw 3245	. . . . . 6 ⊢ (∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑧(Itv‘𝐺)𝑤) ↔ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤))
12	9, 11	bitrdi 290	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤)))
13	6, 12	anbi12d 644	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦)) ↔ ((𝑧 ∈ (𝑃 ∖ 𝐴) ∧ 𝑤 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤))))
14	13	cbvopabv 5184	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))} = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (𝑃 ∖ 𝐴) ∧ 𝑤 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤))}
15		nhpmirhp.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
16		nhpmirhp.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
17		nhpmirhp.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
18		nhpmirhp.e	. . . 4 ⊢ 𝐸 = (hlG‘𝐺)
19		nhpmirhp.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ 𝐴))
20		nhpmirhp.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ((𝐴𝐸𝑌) ∖ 𝐴))
21	20	eldifad 4054	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝐴𝐸𝑌))
22	1, 3, 15, 18, 17, 16, 19, 21	plngssp 29316	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
23		nhpmirhp.s	. . . 4 ⊢ 𝑆 = (pInvG‘𝐺)
24		nhpmirhp.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
25	1, 15, 3, 17, 16, 24	tglnpt 29069	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
26		nhpmirhp.m	. . . 4 ⊢ 𝑀 = (𝑆‘𝑋)
27	1, 2, 3, 15, 23, 17, 25, 26, 22	mircl 29190	. . 3 ⊢ (𝜑 → (𝑀‘𝑍) ∈ 𝑃)
28	20	eldifbd 4055	. . . . 5 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝐴)
29	22, 28	eldifd 4053	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ 𝐴))
30	1, 3, 23, 26, 14, 17, 16, 24, 29, 15	oppmir 29289	. . 3 ⊢ (𝜑 → 𝑍{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))} (𝑀‘𝑍))
31	1, 2, 3, 14, 15, 16, 17, 22, 27, 30	oppcom 29277	. 2 ⊢ (𝜑 → (𝑀‘𝑍){⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))}𝑍)
32	19	eldifad 4054	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
33		nhpmirhp.1	. . . . . 6 ⊢ (𝜑 → ¬ 𝑌((hpG‘𝐺)‘𝐴)𝑍)
34	17	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍((hpG‘𝐺)‘𝐴)𝑌) → 𝐺 ∈ TarskiG)
35	16	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍((hpG‘𝐺)‘𝐴)𝑌) → 𝐴 ∈ ran 𝐿)
36	22	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍((hpG‘𝐺)‘𝐴)𝑌) → 𝑍 ∈ 𝑃)
37	32	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍((hpG‘𝐺)‘𝐴)𝑌) → 𝑌 ∈ 𝑃)
38		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍((hpG‘𝐺)‘𝐴)𝑌) → 𝑍((hpG‘𝐺)‘𝐴)𝑌)
39	1, 3, 15, 34, 35, 36, 14, 37, 38	hpgcom 29302	. . . . . 6 ⊢ ((𝜑 ∧ 𝑍((hpG‘𝐺)‘𝐴)𝑌) → 𝑌((hpG‘𝐺)‘𝐴)𝑍)
40	33, 39	mtand 828	. . . . 5 ⊢ (𝜑 → ¬ 𝑍((hpG‘𝐺)‘𝐴)𝑌)
41	1, 3, 15, 18, 17, 16, 19, 14, 22	elplng 29315	. . . . . 6 ⊢ (𝜑 → (𝑍 ∈ (𝐴𝐸𝑌) ↔ (𝑍 ∈ 𝐴 ∨ 𝑍((hpG‘𝐺)‘𝐴)𝑌 ∨ 𝑍{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))}𝑌)))
42	21, 41	mpbid 235	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ 𝐴 ∨ 𝑍((hpG‘𝐺)‘𝐴)𝑌 ∨ 𝑍{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))}𝑌))
43	28, 40, 42	ecase33d 1505	. . . 4 ⊢ (𝜑 → 𝑍{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))}𝑌)
44	1, 2, 3, 14, 15, 16, 17, 22, 32, 43	oppcom 29277	. . 3 ⊢ (𝜑 → 𝑌{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))}𝑍)
45	1, 3, 15, 14, 17, 16, 32, 27, 22, 44	lnopp2hpgb 29298	. 2 ⊢ (𝜑 → ((𝑀‘𝑍){⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))}𝑍 ↔ 𝑌((hpG‘𝐺)‘𝐴)(𝑀‘𝑍)))
46	31, 45	mpbid 235	1 ⊢ (𝜑 → 𝑌((hpG‘𝐺)‘𝐴)(𝑀‘𝑍))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ∖ cdif 4048   class class class wbr 5108  {copab 5173  ran crn 5659  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  distcds 17481  TarskiGcstrkg 28946  Itvcitv 28952  LineGclng 28953  pInvGcmir 29181  hpGchpg 29292  hlGcplng 29308

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-oadd 8518  df-er 8755  df-map 8887  df-pm 8888  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-dju 10019  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-xnn0 12720  df-z 12734  df-uz 13006  df-fz 13680  df-fzo 13829  df-hash 14516  df-word 14700  df-concat 14757  df-s1 14784  df-s2 15041  df-s3 15042  df-trkgc 28967  df-trkgb 28968  df-trkgcb 28969  df-trkgld 28971  df-trkg 28972  df-cgrg 29031  df-leg 29103  df-hlg 29121  df-mir 29182  df-rag 29226  df-perpg 29228  df-hpg 29293  df-plng 29309

This theorem is used by:  perpeq  29405