Theorem nhpmirhp 29269

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 2761	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
3		eqid 2761	. . 3 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
4		eleq1w 2844	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑃 ∖ 𝐴) ↔ 𝑧 ∈ (𝑃 ∖ 𝐴)))
5		eleq1w 2844	. . . . . 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 7427	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥(Itv‘𝐺)𝑦) = (𝑧(Itv‘𝐺)𝑤))
8	7	eleq2d 2847	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑠 ∈ (𝑧(Itv‘𝐺)𝑤)))
9	8	rexbidv 3187	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑧(Itv‘𝐺)𝑤)))
10		eleq1w 2844	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ (𝑧(Itv‘𝐺)𝑤) ↔ 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤)))
11	10	cbvrexvw 3242	. . . . . 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 29252	. . 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 29005	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
26		nhpmirhp.m	. . . 4 ⊢ 𝑀 = (𝑆‘𝑋)
27	1, 2, 3, 15, 23, 17, 25, 26, 22	mircl 29126	. . 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 29225	. . 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 29213	. 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 29238	. . . . . 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 29251	. . . . . 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 29213	. . 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 29234	. 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 3087   ∖ cdif 3896   class class class wbr 5103  {copab 5167  ran crn 5652  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  distcds 17430  TarskiGcstrkg 28882  Itvcitv 28888  LineGclng 28889  pInvGcmir 29117  hpGchpg 29228  hlGcplng 29244

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-oadd 8473  df-er 8710  df-map 8842  df-pm 8843  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-dju 9975  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-n0 12600  df-xnn0 12673  df-z 12687  df-uz 12959  df-fz 13633  df-fzo 13782  df-hash 14468  df-word 14652  df-concat 14709  df-s1 14736  df-s2 14992  df-s3 14993  df-trkgc 28903  df-trkgb 28904  df-trkgcb 28905  df-trkgld 28907  df-trkg 28908  df-cgrg 28967  df-leg 29039  df-hlg 29057  df-mir 29118  df-rag 29162  df-perpg 29164  df-hpg 29229  df-plng 29245

This theorem is used by:  perpeq  29341