tgextalt2 - Mathbox for Jorge Agra

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Theorem tgextalt2 33077

Description: tgexta 33078's SECOND conjunct, strict, at a supplied second extension point 𝐸. This is NOT a permutation of the first conjunct: 𝐸 is a new point, 𝐶 between 𝐴 and 𝐸, and tgextale 33071 / tgextancgra 33074 are applied a second time at the reconfigured triangle (the roles of 𝐴 and 𝐵 exchanged, exterior angle ⟨“𝐵𝐶𝐸”⟩). tgextaoa 33075 then identifies ⟨“𝐵𝐶𝐸”⟩ with ⟨“𝐴𝐶𝐷”⟩ as vertical angles, and the two halves are transported across that congruence -- leacgra 33064 for ≤_∠, and cgratr 29330 under mtand 828 for the negation, which leacgra 33064 cannot move. 𝐸 is existentially constructed by tgexta 33078, not here. (Contributed by JAG, 12-Aug-2026.)

**Hypotheses**
Ref	Expression
tgextalt2.p	⊢ 𝑃 = (Base‘𝐺)
tgextalt2.m	⊢ − = (dist‘𝐺)
tgextalt2.i	⊢ 𝐼 = (Itv‘𝐺)
tgextalt2.l	⊢ 𝐿 = (LineG‘𝐺)
tgextalt2.k	⊢ 𝐾 = (hlG‘𝐺)
tgextalt2.s	⊢ ≤ = (≤_∠‘𝐺)
tgextalt2.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgextalt2.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgextalt2.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgextalt2.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgextalt2.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgextalt2.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
tgextalt2.ext	⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
tgextalt2.ext2	⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸))
tgextalt2.ncol	⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
tgextalt2.ne	⊢ (𝜑 → 𝐶 ≠ 𝐷)
tgextalt2.ne2	⊢ (𝜑 → 𝐶 ≠ 𝐸)

**Assertion**
Ref	Expression
tgextalt2	⊢ (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩ ∧ ¬ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩))

**Proof of Theorem tgextalt2**
Step	Hyp	Ref	Expression
1		tgextalt2.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		tgextalt2.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
3		tgextalt2.k	. . 3 ⊢ 𝐾 = (hlG‘𝐺)
4		tgextalt2.s	. . 3 ⊢ ≤ = (≤_∠‘𝐺)
5		tgextalt2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		tgextalt2.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
7		tgextalt2.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
8		tgextalt2.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9		tgextalt2.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
10		tgextalt2.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
11		tgextalt2.m	. . . . 5 ⊢ − = (dist‘𝐺)
12		tgextalt2.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
13		tgextalt2.ext2	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸))
14		tgextalt2.ncol	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
15	1, 12, 2, 5, 7, 6, 8, 14	ncolrot2 29026	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
16	1, 12, 2, 5, 8, 7, 6, 15	ncolcom 29024	. . . . . 6 ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
17	1, 12, 2, 5, 7, 8, 6, 16	ncolrot1 29025	. . . . 5 ⊢ (𝜑 → ¬ (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
18		tgextalt2.ne2	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐸)
19	1, 11, 2, 12, 3, 5, 7, 8, 6, 9, 13, 17, 18	tgextale 33071	. . . 4 ⊢ (𝜑 → ⟨“𝐶𝐵𝐴”⟩(≤_∠‘𝐺)⟨“𝐵𝐶𝐸”⟩)
20	4	breqi 5108	. . . 4 ⊢ (⟨“𝐶𝐵𝐴”⟩ ≤ ⟨“𝐵𝐶𝐸”⟩ ↔ ⟨“𝐶𝐵𝐴”⟩(≤_∠‘𝐺)⟨“𝐵𝐶𝐸”⟩)
21	19, 20	sylibr 237	. . 3 ⊢ (𝜑 → ⟨“𝐶𝐵𝐴”⟩ ≤ ⟨“𝐵𝐶𝐸”⟩)
22	1, 12, 2, 5, 7, 6, 8, 14	ncolrot1 29025	. . . . . 6 ⊢ (𝜑 → ¬ (𝐵 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
23	1, 2, 12, 5, 7, 6, 8, 22	ncolne1 29093	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
24	23	necomd 3010	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
25	1, 2, 12, 5, 8, 7, 6, 14	ncolne1 29093	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
26	25	necomd 3010	. . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
27	1, 2, 5, 3, 6, 7, 8, 24, 26	cgraswap 29327	. . 3 ⊢ (𝜑 → ⟨“𝐶𝐵𝐴”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
28		tgextalt2.ext	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
29	1, 2, 12, 5, 8, 7, 6, 14	ncolne2 29094	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
30	29	necomd 3010	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
31		tgextalt2.ne	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐷)
32	31	necomd 3010	. . . 4 ⊢ (𝜑 → 𝐷 ≠ 𝐶)
33	1, 11, 2, 3, 5, 8, 7, 6, 10, 9, 28, 13, 24, 30, 32, 18	tgextaoa 33075	. . 3 ⊢ (𝜑 → ⟨“𝐵𝐶𝐸”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)
34	1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 9, 8, 7, 6, 8, 6, 10, 21, 27, 33	leacgra 33064	. 2 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩)
35	1, 11, 2, 12, 3, 5, 7, 8, 6, 9, 13, 17, 18	tgextancgra 33074	. . 3 ⊢ (𝜑 → ¬ ⟨“𝐶𝐵𝐴”⟩(cgrA‘𝐺)⟨“𝐵𝐶𝐸”⟩)
36	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐺 ∈ TarskiG)
37	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐶 ∈ 𝑃)
38	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐵 ∈ 𝑃)
39	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐴 ∈ 𝑃)
40	10	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐷 ∈ 𝑃)
41	27	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → ⟨“𝐶𝐵𝐴”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
42		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)
43	1, 2, 36, 3, 37, 38, 39, 39, 38, 37, 41, 39, 37, 40, 42	cgratr 29330	. . . 4 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → ⟨“𝐶𝐵𝐴”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)
44	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → 𝐸 ∈ 𝑃)
45	1, 2, 5, 3, 7, 6, 9, 8, 6, 10, 33	cgracom 29329	. . . . 5 ⊢ (𝜑 → ⟨“𝐴𝐶𝐷”⟩(cgrA‘𝐺)⟨“𝐵𝐶𝐸”⟩)
46	45	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → ⟨“𝐴𝐶𝐷”⟩(cgrA‘𝐺)⟨“𝐵𝐶𝐸”⟩)
47	1, 2, 36, 3, 37, 38, 39, 39, 37, 40, 43, 38, 37, 44, 46	cgratr 29330	. . 3 ⊢ ((𝜑 ∧ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) → ⟨“𝐶𝐵𝐴”⟩(cgrA‘𝐺)⟨“𝐵𝐶𝐸”⟩)
48	35, 47	mtand 828	. 2 ⊢ (𝜑 → ¬ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)
49	34, 48	jca 521	1 ⊢ (𝜑 → (⟨“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: tgexta 33078

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