Theorem nhpmirhp 29109

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 2765	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
3		eqid 2765	. . 3 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
4		eleq1w 2848	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑃 ∖ 𝐴) ↔ 𝑧 ∈ (𝑃 ∖ 𝐴)))
5		eleq1w 2848	. . . . . 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 7425	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥(Itv‘𝐺)𝑦) = (𝑧(Itv‘𝐺)𝑤))
8	7	eleq2d 2851	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑠 ∈ (𝑧(Itv‘𝐺)𝑤)))
9	8	rexbidv 3191	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑧(Itv‘𝐺)𝑤)))
10		eleq1w 2848	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ (𝑧(Itv‘𝐺)𝑤) ↔ 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤)))
11	10	cbvrexvw 3246	. . . . . 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 5186	. . 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 3918	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝐴𝐸𝑌))
22	1, 3, 15, 18, 17, 16, 19, 21	plngssp 29092	. . 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 28847	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
26		nhpmirhp.m	. . . 4 ⊢ 𝑀 = (𝑆‘𝑋)
27	1, 2, 3, 15, 23, 17, 25, 26, 22	mircl 28967	. . 3 ⊢ (𝜑 → (𝑀‘𝑍) ∈ 𝑃)
28	20	eldifbd 3919	. . . . 5 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝐴)
29	22, 28	eldifd 3917	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ 𝐴))
30	1, 3, 23, 26, 14, 17, 16, 24, 29, 15	oppmir 29065	. . 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 29054	. 2 ⊢ (𝜑 → (𝑀‘𝑍){⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))}𝑍)
32	19	eldifad 3918	. . 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 29078	. . . . . 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 29091	. . . . . 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 29054	. . 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 29074	. 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 2146  ∃wrex 3091   ∖ cdif 3903   class class class wbr 5111  {copab 5175  ran crn 5664  ‘cfv 6540  (class class class)co 7416  Basecbs 17286  distcds 17336  TarskiGcstrkg 28725  Itvcitv 28731  LineGclng 28732  pInvGcmir 28958  hpGchpg 29068  hlGcplng 29084

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  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 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-dju 9899  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-3 12315  df-n0 12516  df-xnn0 12589  df-z 12603  df-uz 12874  df-fz 13547  df-fzo 13695  df-hash 14380  df-word 14564  df-concat 14621  df-s1 14648  df-s2 14904  df-s3 14905  df-trkgc 28746  df-trkgb 28747  df-trkgcb 28748  df-trkgld 28750  df-trkg 28751  df-cgrg 28809  df-leg 28881  df-hlg 28899  df-mir 28959  df-rag 29003  df-perpg 29005  df-hpg 29069  df-plng 29085

This theorem is used by:  perpeq  29180