Theorem prlngmo 29441

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 28943, in the proof of prlngmolem1 29439. See prlngex 29438 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 7433	. . . . . . 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 5177	. 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 5177	. 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 29440	1 ⊢ (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  ∃*wrmo 3364   ∖ cdif 4044   class class class wbr 5102  {copab 5166  ran crn 5652  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  TarskiGcstrkg 28898  TarskiG_Ecstrkge 28903  Itvcitv 28904  LineGclng 28905  parlnGcprlng 29423

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 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

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 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 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-oadd 8487  df-er 8724  df-map 8856  df-pm 8857  df-en 8981  df-dom 8982  df-sdom 8983  df-fin 8984  df-dju 9989  df-card 10027  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-3 12413  df-n0 12614  df-xnn0 12687  df-z 12701  df-uz 12973  df-fz 13647  df-fzo 13796  df-hash 14482  df-word 14666  df-concat 14723  df-s1 14750  df-s2 15007  df-s3 15008  df-trkgc 28919  df-trkgb 28920  df-trkgcb 28921  df-trkge 28922  df-trkgld 28923  df-trkg 28924  df-cgrg 28983  df-leg 29055  df-hlg 29073  df-mir 29134  df-rag 29178  df-perpg 29180  df-hpg 29245  df-plng 29261  df-prlng 29424

This theorem is used by:  prlngeu  29442  prlngmo2  29443