plngmiropp - Metamath Proof Explorer

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Theorem plngmiropp 29030

Description: Given a line 𝐴 and a point 𝑋 not on 𝐴, then a point 𝑌 on the plane defined by 𝐴 and 𝑋 is either opposite to 𝑋, or opposite to the mirror point of 𝑋 by any point 𝑍 of 𝐴. (Contributed by Thierry Arnoux, 5-Jul-2026.)

**Hypotheses**
Ref	Expression
plngmiropp.p	⊢ 𝑃 = (Base‘𝐺)
plngmiropp.i	⊢ 𝐼 = (Itv‘𝐺)
plngmiropp.l	⊢ 𝐿 = (LineG‘𝐺)
plngmiropp.e	⊢ 𝐸 = (hlG‘𝐺)
plngmiropp.s	⊢ 𝑆 = (pInvG‘𝐺)
plngmiropp.m	⊢ 𝑀 = (𝑆‘𝑍)
plngmiropp.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}
plngmiropp.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
plngmiropp.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
plngmiropp.x	⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))
plngmiropp.y	⊢ (𝜑 → 𝑌 ∈ ((𝐴𝐸𝑋) ∖ 𝐴))
plngmiropp.z	⊢ (𝜑 → 𝑍 ∈ 𝐴)

**Assertion**
Ref	Expression
plngmiropp	⊢ (𝜑 → (𝑋𝑂𝑌 ∨ (𝑀‘𝑋)𝑂𝑌))

Distinct variable groups: 𝐴,𝑎,𝑏,𝑡 𝐺,𝑎,𝑏,𝑡 𝐼,𝑎,𝑏,𝑡 𝐿,𝑎,𝑏,𝑡 𝑀,𝑎,𝑏,𝑡 𝑂,𝑎,𝑏,𝑡 𝑃,𝑎,𝑏,𝑡 𝑋,𝑎,𝑏,𝑡 𝑌,𝑎,𝑏,𝑡 𝑡,𝑍 𝜑,𝑡 Allowed substitution hints: 𝜑(𝑎,𝑏) 𝑆(𝑡,𝑎,𝑏) 𝐸(𝑡,𝑎,𝑏) 𝑍(𝑎,𝑏)

**Proof of Theorem plngmiropp**
Step	Hyp	Ref	Expression
1		plngmiropp.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2769	. . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺)
3		plngmiropp.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		plngmiropp.o	. . . 4 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}
5		plngmiropp.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
6		plngmiropp.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
7	6	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌((hpG‘𝐺)‘𝐴)𝑋) → 𝐴 ∈ ran 𝐿)
8		plngmiropp.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	8	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌((hpG‘𝐺)‘𝐴)𝑋) → 𝐺 ∈ TarskiG)
10		plngmiropp.e	. . . . . 6 ⊢ 𝐸 = (hlG‘𝐺)
11		plngmiropp.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))
12		plngmiropp.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ((𝐴𝐸𝑋) ∖ 𝐴))
13	12	eldifad 3925	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝐴𝐸𝑋))
14	1, 3, 5, 10, 8, 6, 11, 13	plngssp 29017	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
15	14	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌((hpG‘𝐺)‘𝐴)𝑋) → 𝑌 ∈ 𝑃)
16		plngmiropp.s	. . . . . 6 ⊢ 𝑆 = (pInvG‘𝐺)
17		plngmiropp.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝐴)
18	1, 5, 3, 8, 6, 17	tglnpt 28780	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
19		plngmiropp.m	. . . . . 6 ⊢ 𝑀 = (𝑆‘𝑍)
20	11	eldifad 3925	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
21	1, 2, 3, 5, 16, 8, 18, 19, 20	mircl 28896	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑋) ∈ 𝑃)
22	21	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌((hpG‘𝐺)‘𝐴)𝑋) → (𝑀‘𝑋) ∈ 𝑃)
23	20	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌((hpG‘𝐺)‘𝐴)𝑋) → 𝑋 ∈ 𝑃)
24		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌((hpG‘𝐺)‘𝐴)𝑋) → 𝑌((hpG‘𝐺)‘𝐴)𝑋)
25	1, 3, 5, 9, 7, 15, 4, 23, 24	hpgcom 29004	. . . . 5 ⊢ ((𝜑 ∧ 𝑌((hpG‘𝐺)‘𝐴)𝑋) → 𝑋((hpG‘𝐺)‘𝐴)𝑌)
26	1, 3, 16, 19, 4, 8, 6, 17, 11, 5	oppmir 28991	. . . . . . 7 ⊢ (𝜑 → 𝑋𝑂(𝑀‘𝑋))
27	1, 3, 5, 4, 8, 6, 20, 14, 21, 26	lnopp2hpgb 29000	. . . . . 6 ⊢ (𝜑 → (𝑌𝑂(𝑀‘𝑋) ↔ 𝑋((hpG‘𝐺)‘𝐴)𝑌))
28	27	biimpar 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑋((hpG‘𝐺)‘𝐴)𝑌) → 𝑌𝑂(𝑀‘𝑋))
29	25, 28	syldan 602	. . . 4 ⊢ ((𝜑 ∧ 𝑌((hpG‘𝐺)‘𝐴)𝑋) → 𝑌𝑂(𝑀‘𝑋))
30	1, 2, 3, 4, 5, 7, 9, 15, 22, 29	oppcom 28980	. . 3 ⊢ ((𝜑 ∧ 𝑌((hpG‘𝐺)‘𝐴)𝑋) → (𝑀‘𝑋)𝑂𝑌)
31	6	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌𝑂𝑋) → 𝐴 ∈ ran 𝐿)
32	8	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌𝑂𝑋) → 𝐺 ∈ TarskiG)
33	14	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌𝑂𝑋) → 𝑌 ∈ 𝑃)
34	20	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌𝑂𝑋) → 𝑋 ∈ 𝑃)
35		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑌𝑂𝑋) → 𝑌𝑂𝑋)
36	1, 2, 3, 4, 5, 31, 32, 33, 34, 35	oppcom 28980	. . 3 ⊢ ((𝜑 ∧ 𝑌𝑂𝑋) → 𝑋𝑂𝑌)
37	1, 3, 5, 10, 8, 6, 11, 4, 14	elplng 29016	. . . . . 6 ⊢ (𝜑 → (𝑌 ∈ (𝐴𝐸𝑋) ↔ (𝑌 ∈ 𝐴 ∨ 𝑌((hpG‘𝐺)‘𝐴)𝑋 ∨ 𝑌𝑂𝑋)))
38	13, 37	mpbid 235	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ 𝐴 ∨ 𝑌((hpG‘𝐺)‘𝐴)𝑋 ∨ 𝑌𝑂𝑋))
39		3orass 1104	. . . . 5 ⊢ ((𝑌 ∈ 𝐴 ∨ 𝑌((hpG‘𝐺)‘𝐴)𝑋 ∨ 𝑌𝑂𝑋) ↔ (𝑌 ∈ 𝐴 ∨ (𝑌((hpG‘𝐺)‘𝐴)𝑋 ∨ 𝑌𝑂𝑋)))
40	38, 39	sylib 221	. . . 4 ⊢ (𝜑 → (𝑌 ∈ 𝐴 ∨ (𝑌((hpG‘𝐺)‘𝐴)𝑋 ∨ 𝑌𝑂𝑋)))
41	12	eldifbd 3926	. . . 4 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐴)
42	40, 41	orcnd 891	. . 3 ⊢ (𝜑 → (𝑌((hpG‘𝐺)‘𝐴)𝑋 ∨ 𝑌𝑂𝑋))
43	30, 36, 42	orim12da 980	. 2 ⊢ (𝜑 → ((𝑀‘𝑋)𝑂𝑌 ∨ 𝑋𝑂𝑌))
44	43	orcomd 884	1 ⊢ (𝜑 → (𝑋𝑂𝑌 ∨ (𝑀‘𝑋)𝑂𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∨ wo 860 ∨ w3o 1100 = wceq 1567 ∈ wcel 2149 ∃wrex 3095 ∖ cdif 3910 class class class wbr 5110 {copab 5174 ran crn 5660 ‘cfv 6533 (class class class)co 7408 Basecbs 17265 distcds 17315 TarskiGcstrkg 28658 Itvcitv 28664 LineGclng 28665 pInvGcmir 28887 hpGchpg 28994 hlGcplng 29009

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 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 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9883 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-2 12299 df-3 12300 df-n0 12501 df-xnn0 12574 df-z 12588 df-uz 12859 df-fz 13532 df-fzo 13679 df-hash 14363 df-word 14547 df-concat 14604 df-s1 14630 df-s2 14881 df-s3 14882 df-trkgc 28679 df-trkgb 28680 df-trkgcb 28681 df-trkgld 28683 df-trkg 28684 df-cgrg 28742 df-leg 28814 df-hlg 28832 df-mir 28888 df-rag 28929 df-perpg 28931 df-hpg 28995 df-plng 29010

This theorem is referenced by: prlngmolem2 29152

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