Theorem nhpmirhp 29283

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 7431	. . . . . . . 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 5177	. . 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 4050	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝐴𝐸𝑌))
22	1, 3, 15, 18, 17, 16, 19, 21	plngssp 29266	. . 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 29019	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
26		nhpmirhp.m	. . . 4 ⊢ 𝑀 = (𝑆‘𝑋)
27	1, 2, 3, 15, 23, 17, 25, 26, 22	mircl 29140	. . 3 ⊢ (𝜑 → (𝑀‘𝑍) ∈ 𝑃)
28	20	eldifbd 4051	. . . . 5 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝐴)
29	22, 28	eldifd 4049	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ 𝐴))
30	1, 3, 23, 26, 14, 17, 16, 24, 29, 15	oppmir 29239	. . 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 29227	. 2 ⊢ (𝜑 → (𝑀‘𝑍){⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))}𝑍)
32	19	eldifad 4050	. . 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 29252	. . . . . 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 29265	. . . . . 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 29227	. . 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 29248	. 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 1569   ∈ wcel 2144  ∃wrex 3086   ∖ cdif 4044   class class class wbr 5102  {copab 5166  ran crn 5651  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  distcds 17444  TarskiGcstrkg 28896  Itvcitv 28902  LineGclng 28903  pInvGcmir 29131  hpGchpg 29242  hlGcplng 29258

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-oadd 8485  df-er 8722  df-map 8854  df-pm 8855  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-dju 9987  df-card 10025  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-nn 12341  df-2 12410  df-3 12411  df-n0 12612  df-xnn0 12685  df-z 12699  df-uz 12971  df-fz 13645  df-fzo 13794  df-hash 14480  df-word 14664  df-concat 14721  df-s1 14748  df-s2 15005  df-s3 15006  df-trkgc 28917  df-trkgb 28918  df-trkgcb 28919  df-trkgld 28921  df-trkg 28922  df-cgrg 28981  df-leg 29053  df-hlg 29071  df-mir 29132  df-rag 29176  df-perpg 29178  df-hpg 29243  df-plng 29259

This theorem is used by:  perpeq  29355