Theorem prlngmo 29425

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 28927, in the proof of prlngmolem1 29423. See prlngex 29422 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 2844	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑃 ∖ 𝑏) ↔ 𝑧 ∈ (𝑃 ∖ 𝑏)))
9		eleq1w 2844	. . . . 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 2844	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑡 ∈ (𝑥(Itv‘𝐺)𝑦)))
12	11	cbvrexvw 3242	. . . . 5 ⊢ (∃𝑠 ∈ 𝑏 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑡 ∈ 𝑏 𝑡 ∈ (𝑥(Itv‘𝐺)𝑦))
13		oveq12 7427	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥(Itv‘𝐺)𝑦) = (𝑧(Itv‘𝐺)𝑤))
14	13	eleq2d 2847	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑡 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤)))
15	14	rexbidv 3187	. . . . 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 2844	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑃 ∖ 𝐴) ↔ 𝑧 ∈ (𝑃 ∖ 𝐴)))
20		eleq1w 2844	. . . . 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 2844	. . . . . 6 ⊢ (𝑠 = 𝑣 → (𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑣 ∈ (𝑥(Itv‘𝐺)𝑦)))
23	22	cbvrexvw 3242	. . . . 5 ⊢ (∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑣 ∈ 𝐴 𝑣 ∈ (𝑥(Itv‘𝐺)𝑦))
24	13	eleq2d 2847	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑣 ∈ (𝑧(Itv‘𝐺)𝑤)))
25	24	rexbidv 3187	. . . . 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 29424	1 ⊢ (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  ∃*wrmo 3365   ∖ cdif 3896   class class class wbr 5103  {copab 5167  ran crn 5652  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  TarskiGcstrkg 28882  TarskiG_Ecstrkge 28887  Itvcitv 28888  LineGclng 28889  parlnGcprlng 29407

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-trkge 28906  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  df-prlng 29408

This theorem is used by:  prlngeu  29426  prlngmo2  29427