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Theorem tgexta 33108

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 33116's contradiction branch needs, since a bare ≤_∠ conclusion is satisfied by a congruence and so contradicts nothing (tgextan 33109 records that the pair is usable). The second conjunct is not a permutation of the first: axtgsegcon 28956 builds a second extension point 𝑦 beyond 𝐶 on line 𝐴𝐶, tgextalt2 33107 reruns the construction there, and the two exterior angles are identified as vertical by tgextaoa 33105. (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 33106	. 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 2769	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴))) → (𝐶 − 𝐴) = (𝐶 − 𝑦))
38	1, 3, 4, 7, 8, 9, 10, 13	ncolne2 29124	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
39	38	necomd 3014	. . . . . . 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 28975	. . . 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 33107	. . 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 28956	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝐶 ∈ (𝐴𝐼𝑦) ∧ (𝐶 − 𝑦) = (𝐶 − 𝐴)))
45	43, 44	r19.29a 3174	. 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 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: tgextan 33109

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