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Theorem tgexta 33078

Description: The exterior-angle theorem, GeoCoq l11_41 -- BOTH remote interior angles of the triangle are STRICTLY less than the exterior angle at 𝐶. There is no ltA in set.mm , so strictness is carried inline as ≤_∠ together with the non-congruence; the pair is what tgaas 33086's contradiction branch needs, since a bare ≤_∠ conclusion is satisfied by a congruence and so contradicts nothing (tgextan 33079 records that the pair is usable). The second conjunct is not a permutation of the first: axtgsegcon 28926 builds a second extension point 𝑦 beyond 𝐶 on line 𝐴𝐶, tgextalt2 33077 reruns the construction there, and the two exterior angles are identified as vertical by tgextaoa 33075. (Contributed by JAG, 12-Aug-2026.)

**Hypotheses**
Ref	Expression
tgexta.p	⊢ 𝑃 = (Base‘𝐺)
tgexta.m	⊢ − = (dist‘𝐺)
tgexta.i	⊢ 𝐼 = (Itv‘𝐺)
tgexta.l	⊢ 𝐿 = (LineG‘𝐺)
tgexta.k	⊢ 𝐾 = (hlG‘𝐺)
tgexta.s	⊢ ≤ = (≤_∠‘𝐺)
tgexta.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgexta.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgexta.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgexta.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgexta.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgexta.1	⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
tgexta.2	⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
tgexta.3	⊢ (𝜑 → 𝐶 ≠ 𝐷)

**Assertion**
Ref	Expression
tgexta	⊢ (𝜑 → ((⟨“𝐵𝐴𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩ ∧ ¬ ⟨“𝐵𝐴𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) ∧ (⟨“𝐴𝐵𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩ ∧ ¬ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)))

Proof of Theorem tgexta Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgexta.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		tgexta.m	. . 3 ⊢ − = (dist‘𝐺)
3		tgexta.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		tgexta.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
5		tgexta.k	. . 3 ⊢ 𝐾 = (hlG‘𝐺)
6		tgexta.s	. . 3 ⊢ ≤ = (≤_∠‘𝐺)
7		tgexta.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8		tgexta.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9		tgexta.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10		tgexta.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11		tgexta.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
12		tgexta.1	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
13		tgexta.2	. . 3 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
14		tgexta.3	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐷)
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	tgextalt1 33076	. 2 ⊢ (𝜑 → (⟨“𝐵𝐴𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩ ∧ ¬ ⟨“𝐵𝐴𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩))
16	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → 𝐺 ∈ TarskiG)
17	16	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → 𝐺 ∈ TarskiG)
18	8	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → 𝐴 ∈ 𝑃)
19	18	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → 𝐴 ∈ 𝑃)
20	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → 𝐵 ∈ 𝑃)
21	20	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → 𝐵 ∈ 𝑃)
22	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → 𝐶 ∈ 𝑃)
23	22	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → 𝐶 ∈ 𝑃)
24	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → 𝐷 ∈ 𝑃)
25	24	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → 𝐷 ∈ 𝑃)
26		simpl 488	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → (𝜑 ∧ 𝑦 ∈ 𝑃))
27	26	simprd 501	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → 𝑦 ∈ 𝑃)
28	12	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → 𝐶 ∈ (𝐵𝐼𝐷))
29	28	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → 𝐶 ∈ (𝐵𝐼𝐷))
30		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴)))
31	30	simpld 500	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → 𝐶 ∈ (𝐴𝐼𝑦))
32	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
33	32	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
34	14	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → 𝐶 ≠ 𝐷)
35	34	adantr 486	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → 𝐶 ≠ 𝐷)
36	30	simprd 501	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → (𝐶 − 𝑦) = (𝐶 − 𝐴))
37	36	eqcomd 2766	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → (𝐶 − 𝐴) = (𝐶 − 𝑦))
38	1, 3, 4, 7, 8, 9, 10, 13	ncolne2 29094	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
39	38	necomd 3010	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
40	39	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → 𝐶 ≠ 𝐴)
41	40	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → 𝐶 ≠ 𝐴)
42	1, 2, 3, 17, 23, 19, 23, 27, 37, 41	tgcgrneq 28945	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → 𝐶 ≠ 𝑦)
43	1, 2, 3, 4, 5, 6, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 42	tgextalt2 33077	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → (⟨“𝐴𝐵𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩ ∧ ¬ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩))
44	1, 2, 3, 7, 8, 10, 10, 8	axtgsegcon 28926	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴)))
45	43, 44	r19.29a 3170	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩ ∧ ¬ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩))
46	15, 45	jca 521	1 ⊢ (𝜑 → ((⟨“𝐵𝐴𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩ ∧ ¬ ⟨“𝐵𝐴𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) ∧ (⟨“𝐴𝐵𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩ ∧ ¬ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ‘cfv 6538 (class class class)co 7420 ⟨“cs3 14993 Basecbs 17387 distcds 17437 TarskiGcstrkg 28889 Itvcitv 28895 LineGclng 28896 hlGchlg 29063 cgrAccgra 29314 ≤_∠cleag 29355

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 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

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 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 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 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-oadd 8480 df-er 8717 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-n0 12607 df-xnn0 12680 df-z 12694 df-uz 12966 df-fz 13640 df-fzo 13789 df-hash 14475 df-word 14659 df-concat 14716 df-s1 14743 df-s2 14999 df-s3 15000 df-s4 15001 df-trkgc 28910 df-trkgb 28911 df-trkgcb 28912 df-trkgld 28914 df-trkg 28915 df-cgrg 28974 df-ismt 28996 df-leg 29046 df-hlg 29064 df-mir 29125 df-rag 29169 df-perpg 29171 df-hpg 29236 df-mid 29279 df-lmi 29280 df-cgra 29315 df-inag 29356 df-leag 29365

This theorem is used by: tgextan 33079

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