Theorem nhpmirhp 29155

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 2760	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
3		eqid 2760	. . 3 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
4		eleq1w 2843	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑃 ∖ 𝐴) ↔ 𝑧 ∈ (𝑃 ∖ 𝐴)))
5		eleq1w 2843	. . . . . 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 7422	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥(Itv‘𝐺)𝑦) = (𝑧(Itv‘𝐺)𝑤))
8	7	eleq2d 2846	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑠 ∈ (𝑧(Itv‘𝐺)𝑤)))
9	8	rexbidv 3186	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑧(Itv‘𝐺)𝑤)))
10		eleq1w 2843	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ (𝑧(Itv‘𝐺)𝑤) ↔ 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤)))
11	10	cbvrexvw 3241	. . . . . 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 5178	. . 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 3911	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝐴𝐸𝑌))
22	1, 3, 15, 18, 17, 16, 19, 21	plngssp 29138	. . 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 28891	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
26		nhpmirhp.m	. . . 4 ⊢ 𝑀 = (𝑆‘𝑋)
27	1, 2, 3, 15, 23, 17, 25, 26, 22	mircl 29012	. . 3 ⊢ (𝜑 → (𝑀‘𝑍) ∈ 𝑃)
28	20	eldifbd 3912	. . . . 5 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝐴)
29	22, 28	eldifd 3910	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ 𝐴))
30	1, 3, 23, 26, 14, 17, 16, 24, 29, 15	oppmir 29111	. . 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 29099	. 2 ⊢ (𝜑 → (𝑀‘𝑍){⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))}𝑍)
32	19	eldifad 3911	. . 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 29124	. . . . . 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 29137	. . . . . 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 1504	. . . 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 29099	. . 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 29120	. 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 2145  ∃wrex 3086   ∖ cdif 3896   class class class wbr 5103  {copab 5167  ran crn 5656  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  distcds 17351  TarskiGcstrkg 28768  Itvcitv 28774  LineGclng 28775  pInvGcmir 29003  hpGchpg 29114  hlGcplng 29130

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 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

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 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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-er 8696  df-map 8828  df-pm 8829  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-dju 9906  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-n0 12529  df-xnn0 12602  df-z 12616  df-uz 12888  df-fz 13562  df-fzo 13710  df-hash 14395  df-word 14579  df-concat 14636  df-s1 14663  df-s2 14919  df-s3 14920  df-trkgc 28789  df-trkgb 28790  df-trkgcb 28791  df-trkgld 28793  df-trkg 28794  df-cgrg 28853  df-leg 28925  df-hlg 28943  df-mir 29004  df-rag 29048  df-perpg 29050  df-hpg 29115  df-plng 29131

This theorem is used by:  perpeq  29227