Theorem prlngmo 29153

Description: Playfair's axiom. Given a line 𝐴 and a point 𝑋 not on 𝐴, at most one line parallel to 𝐴 can be drawn through 𝑋. Theorem 12.11 of [Schwabhauser] p. 123. Note that this is the first instance of a theorem where the geometry is required to be Euclidean, as expressed by 𝐺 ∈ TarskiG_E. Theorem A10 of [Schwabhauser] p. 24 is used, in the form of axtgeucl 28703, in the proof of prlngmolem1 29151. See prlngex 29150 for the corresponding existence theorem. (Contributed by Thierry Arnoux, 5-Jul-2026.)

Hypotheses

Ref	Expression
prlngeu.p	⊢ 𝑃 = (Base‘𝐺)
prlngeu.l	⊢ 𝐿 = (LineG‘𝐺)
prlngeu.r	⊢ ∥ = (parlnG‘𝐺)
prlngeu.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
prlngeu.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
prlngeu.x	⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))
prlngeu.1	⊢ (𝜑 → 𝐺 ∈ TarskiGE)

Assertion

Ref	Expression
prlngmo	⊢ (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))

Distinct variable groups:   ∥ ,𝑏   𝐴,𝑏   𝐿,𝑏   𝑋,𝑏   𝜑,𝑏

Allowed substitution hints:   𝑃(𝑏)   𝐺(𝑏)

Proof of Theorem prlngmo

Dummy variables 𝑠 𝑡 𝑤 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prlngeu.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		prlngeu.l	. 2 ⊢ 𝐿 = (LineG‘𝐺)
3		prlngeu.r	. 2 ⊢ ∥ = (parlnG‘𝐺)
4		prlngeu.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		prlngeu.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
6		prlngeu.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))
7		prlngeu.1	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiGE)
8		eleq1w 2852	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑃 ∖ 𝑏) ↔ 𝑧 ∈ (𝑃 ∖ 𝑏)))
9		eleq1w 2852	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝑃 ∖ 𝑏) ↔ 𝑤 ∈ (𝑃 ∖ 𝑏)))
10	8, 9	bi2anan9 649	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (𝑃 ∖ 𝑏) ∧ 𝑦 ∈ (𝑃 ∖ 𝑏)) ↔ (𝑧 ∈ (𝑃 ∖ 𝑏) ∧ 𝑤 ∈ (𝑃 ∖ 𝑏))))
11		eleq1w 2852	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑡 ∈ (𝑥(Itv‘𝐺)𝑦)))
12	11	cbvrexvw 3250	. . . . 5 ⊢ (∃𝑠 ∈ 𝑏 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑡 ∈ 𝑏 𝑡 ∈ (𝑥(Itv‘𝐺)𝑦))
13		oveq12 7417	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥(Itv‘𝐺)𝑦) = (𝑧(Itv‘𝐺)𝑤))
14	13	eleq2d 2855	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑡 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤)))
15	14	rexbidv 3195	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑡 ∈ 𝑏 𝑡 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑡 ∈ 𝑏 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤)))
16	12, 15	bitrid 286	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑠 ∈ 𝑏 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑡 ∈ 𝑏 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤)))
17	10, 16	anbi12d 643	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ (𝑃 ∖ 𝑏) ∧ 𝑦 ∈ (𝑃 ∖ 𝑏)) ∧ ∃𝑠 ∈ 𝑏 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦)) ↔ ((𝑧 ∈ (𝑃 ∖ 𝑏) ∧ 𝑤 ∈ (𝑃 ∖ 𝑏)) ∧ ∃𝑡 ∈ 𝑏 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤))))
18	17	cbvopabv 5185	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝑏) ∧ 𝑦 ∈ (𝑃 ∖ 𝑏)) ∧ ∃𝑠 ∈ 𝑏 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))} = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (𝑃 ∖ 𝑏) ∧ 𝑤 ∈ (𝑃 ∖ 𝑏)) ∧ ∃𝑡 ∈ 𝑏 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤))}
19		eleq1w 2852	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑃 ∖ 𝐴) ↔ 𝑧 ∈ (𝑃 ∖ 𝐴)))
20		eleq1w 2852	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝑃 ∖ 𝐴) ↔ 𝑤 ∈ (𝑃 ∖ 𝐴)))
21	19, 20	bi2anan9 649	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ↔ (𝑧 ∈ (𝑃 ∖ 𝐴) ∧ 𝑤 ∈ (𝑃 ∖ 𝐴))))
22		eleq1w 2852	. . . . . 6 ⊢ (𝑠 = 𝑣 → (𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑣 ∈ (𝑥(Itv‘𝐺)𝑦)))
23	22	cbvrexvw 3250	. . . . 5 ⊢ (∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑣 ∈ 𝐴 𝑣 ∈ (𝑥(Itv‘𝐺)𝑦))
24	13	eleq2d 2855	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑣 ∈ (𝑧(Itv‘𝐺)𝑤)))
25	24	rexbidv 3195	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑣 ∈ 𝐴 𝑣 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑣 ∈ 𝐴 𝑣 ∈ (𝑧(Itv‘𝐺)𝑤)))
26	23, 25	bitrid 286	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑣 ∈ 𝐴 𝑣 ∈ (𝑧(Itv‘𝐺)𝑤)))
27	21, 26	anbi12d 643	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦)) ↔ ((𝑧 ∈ (𝑃 ∖ 𝐴) ∧ 𝑤 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑣 ∈ 𝐴 𝑣 ∈ (𝑧(Itv‘𝐺)𝑤))))
28	27	cbvopabv 5185	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))} = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (𝑃 ∖ 𝐴) ∧ 𝑤 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑣 ∈ 𝐴 𝑣 ∈ (𝑧(Itv‘𝐺)𝑤))}
29	1, 2, 3, 4, 5, 6, 7, 18, 28	prlngmolem2 29152	1 ⊢ (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  ∃*wrmo 3375   ∖ cdif 3910   class class class wbr 5110  {copab 5174  ran crn 5660  ‘cfv 6533  (class class class)co 7408  Basecbs 17265  TarskiGcstrkg 28658  TarskiG_Ecstrkge 28663  Itvcitv 28664  LineGclng 28665  parlnGcprlng 29137

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-trkge 28682  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  df-prlng 29138

This theorem is referenced by:  prlngeu  29154