Theorem prlngmo 29311

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 28813, in the proof of prlngmolem1 29309. See prlngex 29308 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 2843	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑃 ∖ 𝑏) ↔ 𝑧 ∈ (𝑃 ∖ 𝑏)))
9		eleq1w 2843	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝑃 ∖ 𝑏) ↔ 𝑤 ∈ (𝑃 ∖ 𝑏)))
10	8, 9	bi2anan9 650	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (𝑃 ∖ 𝑏) ∧ 𝑦 ∈ (𝑃 ∖ 𝑏)) ↔ (𝑧 ∈ (𝑃 ∖ 𝑏) ∧ 𝑤 ∈ (𝑃 ∖ 𝑏))))
11		eleq1w 2843	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑡 ∈ (𝑥(Itv‘𝐺)𝑦)))
12	11	cbvrexvw 3241	. . . . 5 ⊢ (∃𝑠 ∈ 𝑏 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑡 ∈ 𝑏 𝑡 ∈ (𝑥(Itv‘𝐺)𝑦))
13		oveq12 7422	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥(Itv‘𝐺)𝑦) = (𝑧(Itv‘𝐺)𝑤))
14	13	eleq2d 2846	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑡 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤)))
15	14	rexbidv 3186	. . . . 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 644	. . 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 5178	. 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 2843	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑃 ∖ 𝐴) ↔ 𝑧 ∈ (𝑃 ∖ 𝐴)))
20		eleq1w 2843	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (𝑃 ∖ 𝐴) ↔ 𝑤 ∈ (𝑃 ∖ 𝐴)))
21	19, 20	bi2anan9 650	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ↔ (𝑧 ∈ (𝑃 ∖ 𝐴) ∧ 𝑤 ∈ (𝑃 ∖ 𝐴))))
22		eleq1w 2843	. . . . . 6 ⊢ (𝑠 = 𝑣 → (𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑣 ∈ (𝑥(Itv‘𝐺)𝑦)))
23	22	cbvrexvw 3241	. . . . 5 ⊢ (∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑣 ∈ 𝐴 𝑣 ∈ (𝑥(Itv‘𝐺)𝑦))
24	13	eleq2d 2846	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑣 ∈ (𝑧(Itv‘𝐺)𝑤)))
25	24	rexbidv 3186	. . . . 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 644	. . 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 5178	. 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 29310	1 ⊢ (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  ∃*wrmo 3364   ∖ cdif 3896   class class class wbr 5103  {copab 5167  ran crn 5656  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  TarskiGcstrkg 28768  TarskiG_Ecstrkge 28773  Itvcitv 28774  LineGclng 28775  parlnGcprlng 29293

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

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 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 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 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  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 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  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 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-dju 9906  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-n0 12529  df-xnn0 12602  df-z 12616  df-uz 12888  df-fz 13562  df-fzo 13710  df-hash 14395  df-word 14579  df-concat 14636  df-s1 14663  df-s2 14919  df-s3 14920  df-trkgc 28789  df-trkgb 28790  df-trkgcb 28791  df-trkge 28792  df-trkgld 28793  df-trkg 28794  df-cgrg 28853  df-leg 28925  df-hlg 28943  df-mir 29004  df-rag 29048  df-perpg 29050  df-hpg 29115  df-plng 29131  df-prlng 29294

This theorem is used by:  prlngeu  29312  prlngmo2  29313