Theorem tgaasexc 33111

Description: The contradiction branch of AAS, stated once for both orders of the two candidate points. 𝑆 lies strictly between 𝐸 and 𝑇, and 𝐷 is off the line, so ⟨“𝐷𝑆𝐸”⟩ is an exterior angle of triangle 𝐷𝑇𝑆 and is STRICTLY greater than the remote interior angle at 𝑇-- while the hypothesis says the two are congruent. tgextan 33109 turns that into anything. (Contributed by JAG, 12-Aug-2026.)

Hypotheses

Ref	Expression
tgaasexc.p	⊢ 𝑃 = (Base‘𝐺)
tgaasexc.m	⊢ − = (dist‘𝐺)
tgaasexc.i	⊢ 𝐼 = (Itv‘𝐺)
tgaasexc.l	⊢ 𝐿 = (LineG‘𝐺)
tgaasexc.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgaasexc.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgaasexc.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
tgaasexc.s	⊢ (𝜑 → 𝑆 ∈ 𝑃)
tgaasexc.t	⊢ (𝜑 → 𝑇 ∈ 𝑃)
tgaasexc.1	⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝑇) ∨ 𝐸 = 𝑇))
tgaasexc.2	⊢ (𝜑 → 𝑆 ∈ (𝐸𝐼𝑇))
tgaasexc.3	⊢ (𝜑 → 𝑆 ≠ 𝐸)
tgaasexc.4	⊢ (𝜑 → 𝑆 ≠ 𝑇)
tgaasexc.5	⊢ (𝜑 → ⟨“𝐷𝑇𝐸”⟩(cgrA‘𝐺)⟨“𝐷𝑆𝐸”⟩)

Assertion

Ref	Expression
tgaasexc	⊢ (𝜑 → 𝜓)

Proof of Theorem tgaasexc

Step	Hyp	Ref	Expression
1		tgaasexc.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		tgaasexc.m	. 2 ⊢ − = (dist‘𝐺)
3		tgaasexc.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		tgaasexc.l	. 2 ⊢ 𝐿 = (LineG‘𝐺)
5		eqid 2763	. 2 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
6		eqid 2763	. 2 ⊢ (≤∠‘𝐺) = (≤∠‘𝐺)
7		tgaasexc.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8		tgaasexc.d	. 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
9		tgaasexc.t	. 2 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
10		tgaasexc.s	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
11		tgaasexc.e	. 2 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
12		tgaasexc.2	. . 3 ⊢ (𝜑 → 𝑆 ∈ (𝐸𝐼𝑇))
13	1, 2, 3, 7, 11, 10, 9, 12	tgbtwncom 28981	. 2 ⊢ (𝜑 → 𝑆 ∈ (𝑇𝐼𝐸))
14		tgaasexc.1	. . 3 ⊢ (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝑇) ∨ 𝐸 = 𝑇))
15	1, 4, 3, 7, 9, 11, 10, 13	btwncolg1 29048	. . 3 ⊢ (𝜑 → (𝑆 ∈ (𝑇𝐿𝐸) ∨ 𝑇 = 𝐸))
16		tgaasexc.4	. . 3 ⊢ (𝜑 → 𝑆 ≠ 𝑇)
17	1, 3, 4, 7, 8, 11, 9, 10, 14, 15, 16	tgaasncol 33110	. 2 ⊢ (𝜑 → ¬ (𝐷 ∈ (𝑇𝐿𝑆) ∨ 𝑇 = 𝑆))
18		tgaasexc.3	. 2 ⊢ (𝜑 → 𝑆 ≠ 𝐸)
19	1, 3, 4, 7, 8, 11, 9, 14	ncolne2 29124	. . . . . 6 ⊢ (𝜑 → 𝐷 ≠ 𝑇)
20	1, 4, 3, 7, 11, 9, 8, 14	ncolrot1 29055	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝐸 ∈ (𝑇𝐿𝐷) ∨ 𝑇 = 𝐷))
21	1, 3, 4, 7, 11, 9, 8, 20	ncolne1 29123	. . . . . . 7 ⊢ (𝜑 → 𝐸 ≠ 𝑇)
22	21	necomd 3014	. . . . . 6 ⊢ (𝜑 → 𝑇 ≠ 𝐸)
23	1, 3, 7, 5, 8, 9, 11, 19, 22	cgraid 29356	. . . . 5 ⊢ (𝜑 → ⟨“𝐷𝑇𝐸”⟩(cgrA‘𝐺)⟨“𝐷𝑇𝐸”⟩)
24	1, 3, 5, 9, 11, 10, 7, 8, 13, 22, 16	btwnhl1 29108	. . . . 5 ⊢ (𝜑 → 𝑆((hlG‘𝐺)‘𝑇)𝐸)
25	1, 3, 5, 7, 8, 9, 11, 8, 9, 11, 23, 10, 24	cgrahl2 29354	. . . 4 ⊢ (𝜑 → ⟨“𝐷𝑇𝐸”⟩(cgrA‘𝐺)⟨“𝐷𝑇𝑆”⟩)
26	1, 3, 7, 5, 8, 9, 11, 8, 9, 10, 25	cgracom 29359	. . 3 ⊢ (𝜑 → ⟨“𝐷𝑇𝑆”⟩(cgrA‘𝐺)⟨“𝐷𝑇𝐸”⟩)
27		tgaasexc.5	. . 3 ⊢ (𝜑 → ⟨“𝐷𝑇𝐸”⟩(cgrA‘𝐺)⟨“𝐷𝑆𝐸”⟩)
28	1, 3, 7, 5, 8, 9, 10, 8, 9, 11, 26, 8, 10, 11, 27	cgratr 29360	. 2 ⊢ (𝜑 → ⟨“𝐷𝑇𝑆”⟩(cgrA‘𝐺)⟨“𝐷𝑆𝐸”⟩)
29	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 17, 18, 28	tgextan 33109	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  ⟨“cs3 15021  Basecbs 17416  distcds 17466  TarskiGcstrkg 28919  Itvcitv 28925  LineGclng 28926  hlGchlg 29093  cgrAccgra 29344  ≤_∠cleag 29385

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-oadd 8508  df-er 8745  df-map 8877  df-pm 8878  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-dju 10009  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-n0 12634  df-xnn0 12707  df-z 12721  df-uz 12993  df-fz 13667  df-fzo 13816  df-hash 14502  df-word 14686  df-concat 14743  df-s1 14770  df-s2 15027  df-s3 15028  df-s4 15029  df-trkgc 28940  df-trkgb 28941  df-trkgcb 28942  df-trkgld 28944  df-trkg 28945  df-cgrg 29004  df-ismt 29026  df-leg 29076  df-hlg 29094  df-mir 29155  df-rag 29199  df-perpg 29201  df-hpg 29266  df-mid 29309  df-lmi 29310  df-cgra 29345  df-inag 29386  df-leag 29395

This theorem is used by:  tgaasne  33112