Theorem prlngmo 29185

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 28722, in the proof of prlngmolem1 29183. See prlngex 29182 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 2846	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑃 ∖ 𝑏) ↔ 𝑧 ∈ (𝑃 ∖ 𝑏)))
9		eleq1w 2846	. . . . 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 2846	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑡 ∈ (𝑥(Itv‘𝐺)𝑦)))
12	11	cbvrexvw 3244	. . . . 5 ⊢ (∃𝑠 ∈ 𝑏 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑡 ∈ 𝑏 𝑡 ∈ (𝑥(Itv‘𝐺)𝑦))
13		oveq12 7421	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥(Itv‘𝐺)𝑦) = (𝑧(Itv‘𝐺)𝑤))
14	13	eleq2d 2849	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑡 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑡 ∈ (𝑧(Itv‘𝐺)𝑤)))
15	14	rexbidv 3189	. . . . 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 2846	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝑃 ∖ 𝐴) ↔ 𝑧 ∈ (𝑃 ∖ 𝐴)))
20		eleq1w 2846	. . . . 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 2846	. . . . . 6 ⊢ (𝑠 = 𝑣 → (𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑣 ∈ (𝑥(Itv‘𝐺)𝑦)))
23	22	cbvrexvw 3244	. . . . 5 ⊢ (∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ ∃𝑣 ∈ 𝐴 𝑣 ∈ (𝑥(Itv‘𝐺)𝑦))
24	13	eleq2d 2849	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑣 ∈ (𝑥(Itv‘𝐺)𝑦) ↔ 𝑣 ∈ (𝑧(Itv‘𝐺)𝑤)))
25	24	rexbidv 3189	. . . . 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 29184	1 ⊢ (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  ∃*wrmo 3368   ∖ cdif 3903   class class class wbr 5110  {copab 5174  ran crn 5664  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  TarskiGcstrkg 28677  TarskiG_Ecstrkge 28682  Itvcitv 28683  LineGclng 28684  parlnGcprlng 29167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-oadd 8458  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-dju 9888  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-n0 12506  df-xnn0 12579  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-hash 14369  df-word 14553  df-concat 14610  df-s1 14636  df-s2 14887  df-s3 14888  df-trkgc 28698  df-trkgb 28699  df-trkgcb 28700  df-trkge 28701  df-trkgld 28702  df-trkg 28703  df-cgrg 28761  df-leg 28833  df-hlg 28851  df-mir 28911  df-rag 28955  df-perpg 28957  df-hpg 29021  df-plng 29037  df-prlng 29168

This theorem is referenced by:  prlngeu  29186  prlngmo2  29187