tgextalt2 - Mathbox for Jorge Agra

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Theorem tgextalt2 33107

Description: tgexta 33108'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 33101 / tgextancgra 33104 are applied a second time at the reconfigured triangle (the roles of 𝐴 and 𝐵 exchanged, exterior angle ⟨“𝐵𝐶𝐸”⟩). tgextaoa 33105 then identifies ⟨“𝐵𝐶𝐸”⟩ with ⟨“𝐴𝐶𝐷”⟩ as vertical angles, and the two halves are transported across that congruence -- leacgra 33094 for ≤_∠, and cgratr 29360 under mtand 828 for the negation, which leacgra 33094 cannot move. 𝐸 is existentially constructed by tgexta 33108, 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 29056	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
16	1, 12, 2, 5, 8, 7, 6, 15	ncolcom 29054	. . . . . 6 ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
17	1, 12, 2, 5, 7, 8, 6, 16	ncolrot1 29055	. . . . 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 33101	. . . 4 ⊢ (𝜑 → ⟨“𝐶𝐵𝐴”⟩(≤_∠‘𝐺)⟨“𝐵𝐶𝐸”⟩)
20	4	breqi 5112	. . . 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 29055	. . . . . 6 ⊢ (𝜑 → ¬ (𝐵 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
23	1, 2, 12, 5, 7, 6, 8, 22	ncolne1 29123	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
24	23	necomd 3014	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
25	1, 2, 12, 5, 8, 7, 6, 14	ncolne1 29123	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
26	25	necomd 3014	. . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
27	1, 2, 5, 3, 6, 7, 8, 24, 26	cgraswap 29357	. . 3 ⊢ (𝜑 → ⟨“𝐶𝐵𝐴”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
28		tgextalt2.ext	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
29	1, 2, 12, 5, 8, 7, 6, 14	ncolne2 29124	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
30	29	necomd 3014	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
31		tgextalt2.ne	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐷)
32	31	necomd 3014	. . . 4 ⊢ (𝜑 → 𝐷 ≠ 𝐶)
33	1, 11, 2, 3, 5, 8, 7, 6, 10, 9, 28, 13, 24, 30, 32, 18	tgextaoa 33105	. . 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 33094	. 2 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩)
35	1, 11, 2, 12, 3, 5, 7, 8, 6, 9, 13, 17, 18	tgextancgra 33104	. . 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 29360	. . . 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 29359	. . . . 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 29360	. . 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 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: tgexta 33108

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