Theorem tgaaddcpbllem2 29204

Description: Lemma for tgaaddcpbl 29206. (Contributed by Thierry Arnoux, 2-Aug-2026.)

Hypotheses

Ref	Expression
tgaaddcpbl.p	⊢ 𝑃 = (Base‘𝐺)
tgaaddcpbl.i	⊢ 𝐼 = (Itv‘𝐺)
tgaaddcpbl.l	⊢ 𝐿 = (LineG‘𝐺)
tgaaddcpbl.c	⊢ ∼ = (cgrA‘𝐺)
tgaaddcpbl.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑆)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑆))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑆)𝑠 ∈ (𝑎𝐼𝑏))}
tgaaddcpbl.q	⊢ 𝑄 = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑))}
tgaaddcpbl.1	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgaaddcpbl.s	⊢ (𝜑 → 𝑆 ∈ 𝑃)
tgaaddcpbl.t	⊢ (𝜑 → 𝑇 ∈ 𝑃)
tgaaddcpbl.u	⊢ (𝜑 → 𝑈 ∈ 𝑃)
tgaaddcpbl.v	⊢ (𝜑 → 𝑉 ∈ 𝑃)
tgaaddcpbl.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
tgaaddcpbl.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tgaaddcpbl.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tgaaddcpbl.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
tgaaddcpbl.2	⊢ (𝜑 → 𝑌 ≠ 𝑆)
tgaaddcpbl.3	⊢ (𝜑 → 𝑉 ≠ 𝑇)
tgaaddcpbl.4	⊢ (𝜑 → 𝑋𝑂𝑍)
tgaaddcpbl.5	⊢ (𝜑 → 𝑈𝑄𝑊)
tgaaddcpbl.6	⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)
tgaaddcpbl.7	⊢ (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)
tgaaddcpbllem3.1	⊢ (𝜑 → ¬ 𝑌 ∈ (𝑋𝐼𝑍))
tgaaddcpbllem2.1	⊢ (𝜑 → 𝑅 ∈ (𝑌𝐿𝑆))
tgaaddcpbllem2.2	⊢ (𝜑 → 𝑅 ∈ (𝑋𝐼𝑍))
tgaaddcpbllem2.3	⊢ (𝜑 → 𝑌 ∈ (𝑆𝐼𝑅))
tgaaddcpbllem2.m	⊢ 𝑀 = ((pInvG‘𝐺)‘𝑉)
tgaaddcpbllem2.k	⊢ 𝐾 = (hlG‘𝐺)

Assertion

Ref	Expression
tgaaddcpbllem2	⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)

Distinct variable groups:   ∼ ,𝑠   𝑡, ∼   𝐺,𝑐,𝑑,𝑡   𝐺,𝑠   𝐼,𝑎,𝑏,𝑠   𝐼,𝑐,𝑑,𝑡   𝑡,𝐾   𝐿,𝑎,𝑏,𝑠   𝐿,𝑐,𝑑,𝑡   𝑂,𝑠   𝑃,𝑎,𝑏,𝑠   𝑃,𝑐,𝑑,𝑡   𝑄,𝑐,𝑑,𝑡   𝑡,𝑅   𝑆,𝑎,𝑏,𝑠   𝑡,𝑆   𝑇,𝑐,𝑑,𝑡   𝑈,𝑐,𝑑,𝑡   𝑈,𝑠   𝑉,𝑐,𝑑,𝑡   𝑉,𝑠   𝑊,𝑠   𝑡,𝑊   𝑋,𝑠   𝑡,𝑋   𝑌,𝑎,𝑏,𝑠   𝑡,𝑌   𝑍,𝑠   𝑡,𝑍   𝜑,𝑠   𝜑,𝑡   𝑎,𝑐,𝑑,𝑡,𝑏,𝑀   𝑅,𝑎,𝑏,𝑠   𝑇,𝑎,𝑏   𝑉,𝑎,𝑏   𝜑,𝑐,𝑑

Allowed substitution hints:   𝜑(𝑎, 𝑏)   𝑄(𝑠, 𝑎, 𝑏)   ∼ (𝑎, 𝑏, 𝑐, 𝑑)   𝑅(𝑐, 𝑑)   𝑆(𝑐, 𝑑)   𝑇(𝑠)   𝑈(𝑎, 𝑏)   𝐺(𝑎, 𝑏)   𝐾(𝑠, 𝑎, 𝑏, 𝑐, 𝑑)   𝑀(𝑠)   𝑂(𝑡, 𝑎, 𝑏, 𝑐, 𝑑)   𝑊(𝑎, 𝑏, 𝑐, 𝑑)   𝑋(𝑎, 𝑏, 𝑐, 𝑑)   𝑌(𝑐, 𝑑)   𝑍(𝑎, 𝑏, 𝑐, 𝑑)

Proof of Theorem tgaaddcpbllem2

Dummy variables 𝑓 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgaaddcpbl.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		tgaaddcpbl.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
3		tgaaddcpbl.l	. 2 ⊢ 𝐿 = (LineG‘𝐺)
4		tgaaddcpbl.c	. 2 ⊢ ∼ = (cgrA‘𝐺)
5		eleq1w 2848	. . . . 5 ⊢ (𝑒 = 𝑠 → (𝑒 ∈ (𝑎𝐼𝑏) ↔ 𝑠 ∈ (𝑎𝐼𝑏)))
6	5	cbvrexvw 3246	. . . 4 ⊢ (∃𝑒 ∈ (𝑌𝐿𝑅)𝑒 ∈ (𝑎𝐼𝑏) ↔ ∃𝑠 ∈ (𝑌𝐿𝑅)𝑠 ∈ (𝑎𝐼𝑏))
7	6	anbi2i 635	. . 3 ⊢ (((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑅)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑅))) ∧ ∃𝑒 ∈ (𝑌𝐿𝑅)𝑒 ∈ (𝑎𝐼𝑏)) ↔ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑅)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑅))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑅)𝑠 ∈ (𝑎𝐼𝑏)))
8	7	opabbii 5180	. 2 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑅)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑅))) ∧ ∃𝑒 ∈ (𝑌𝐿𝑅)𝑒 ∈ (𝑎𝐼𝑏))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑅)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑅))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑅)𝑠 ∈ (𝑎𝐼𝑏))}
9		eleq1w 2848	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ↔ 𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))))
10		eleq1w 2848	. . . . 5 ⊢ (𝑏 = 𝑑 → (𝑏 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ↔ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))))
11	9, 10	bi2anan9 650	. . . 4 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → ((𝑎 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ∧ 𝑏 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))) ↔ (𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))))))
12		oveq12 7428	. . . . . . 7 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑎𝐼𝑏) = (𝑐𝐼𝑑))
13	12	eleq2d 2851	. . . . . 6 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑓 ∈ (𝑎𝐼𝑏) ↔ 𝑓 ∈ (𝑐𝐼𝑑)))
14	13	rexbidv 3191	. . . . 5 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑎𝐼𝑏) ↔ ∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑐𝐼𝑑)))
15		eleq1w 2848	. . . . . 6 ⊢ (𝑓 = 𝑡 → (𝑓 ∈ (𝑐𝐼𝑑) ↔ 𝑡 ∈ (𝑐𝐼𝑑)))
16	15	cbvrexvw 3246	. . . . 5 ⊢ (∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑐𝐼𝑑) ↔ ∃𝑡 ∈ (𝑉𝐿(𝑀‘𝑇))𝑡 ∈ (𝑐𝐼𝑑))
17	14, 16	bitrdi 290	. . . 4 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑎𝐼𝑏) ↔ ∃𝑡 ∈ (𝑉𝐿(𝑀‘𝑇))𝑡 ∈ (𝑐𝐼𝑑)))
18	11, 17	anbi12d 644	. . 3 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (((𝑎 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ∧ 𝑏 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))) ∧ ∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑎𝐼𝑏)) ↔ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))) ∧ ∃𝑡 ∈ (𝑉𝐿(𝑀‘𝑇))𝑡 ∈ (𝑐𝐼𝑑))))
19	18	cbvopabv 5186	. 2 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ∧ 𝑏 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))) ∧ ∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑎𝐼𝑏))} = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))) ∧ ∃𝑡 ∈ (𝑉𝐿(𝑀‘𝑇))𝑡 ∈ (𝑐𝐼𝑑))}
20		tgaaddcpbl.1	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
21		tgaaddcpbl.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
22		tgaaddcpbl.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
23		tgaaddcpbl.2	. . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑆)
24	1, 2, 3, 20, 21, 22, 23	tgelrnln 28954	. . 3 ⊢ (𝜑 → (𝑌𝐿𝑆) ∈ ran 𝐿)
25		tgaaddcpbllem2.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐿𝑆))
26	1, 3, 2, 20, 24, 25	tglnpt 28869	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
27		eqid 2765	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
28		eqid 2765	. . 3 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
29		tgaaddcpbl.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ 𝑃)
30		tgaaddcpbllem2.m	. . 3 ⊢ 𝑀 = ((pInvG‘𝐺)‘𝑉)
31		tgaaddcpbl.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
32	1, 27, 2, 3, 28, 20, 29, 30, 31	mircl 28989	. 2 ⊢ (𝜑 → (𝑀‘𝑇) ∈ 𝑃)
33		tgaaddcpbl.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
34		tgaaddcpbl.w	. 2 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
35		tgaaddcpbl.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
36		tgaaddcpbl.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
37		tgaaddcpbllem2.2	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐼𝑍))
38		tgaaddcpbllem3.1	. . . . 5 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑋𝐼𝑍))
39	37, 38	elnelneq2d 3060	. . . 4 ⊢ (𝜑 → ¬ 𝑅 = 𝑌)
40	39	neqned 2967	. . 3 ⊢ (𝜑 → 𝑅 ≠ 𝑌)
41	40	necomd 3015	. 2 ⊢ (𝜑 → 𝑌 ≠ 𝑅)
42		tgaaddcpbl.3	. . . . 5 ⊢ (𝜑 → 𝑉 ≠ 𝑇)
43	42	necomd 3015	. . . 4 ⊢ (𝜑 → 𝑇 ≠ 𝑉)
44	1, 27, 2, 3, 28, 20, 29, 30, 31, 43	mirne 28995	. . 3 ⊢ (𝜑 → (𝑀‘𝑇) ≠ 𝑉)
45	44	necomd 3015	. 2 ⊢ (𝜑 → 𝑉 ≠ (𝑀‘𝑇))
46	1, 2, 3, 20, 21, 26, 41	tglinerflx2 28958	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐿𝑅))
47		tgaaddcpbl.o	. . . . 5 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑆)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑆))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑆)𝑠 ∈ (𝑎𝐼𝑏))}
48		tgaaddcpbl.4	. . . . 5 ⊢ (𝜑 → 𝑋𝑂𝑍)
49	1, 27, 2, 47, 3, 24, 20, 35, 36, 48	oppne1 29073	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
50	1, 2, 3, 20, 21, 22, 23, 26, 40, 25	tglineelsb2 28956	. . . 4 ⊢ (𝜑 → (𝑌𝐿𝑆) = (𝑌𝐿𝑅))
51	49, 50	neleqtrd 2887	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑅))
52	1, 27, 2, 47, 3, 24, 20, 35, 36, 48	oppne2 29074	. . . 4 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑆))
53	52, 50	neleqtrd 2887	. . 3 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑅))
54	1, 27, 2, 8, 35, 36, 46, 51, 53, 37	islnoppd 29072	. 2 ⊢ (𝜑 → 𝑋{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑅)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑅))) ∧ ∃𝑒 ∈ (𝑌𝐿𝑅)𝑒 ∈ (𝑎𝐼𝑏))}𝑍)
55	1, 27, 2, 3, 28, 20, 29, 30, 31	mirbtwn 28986	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ ((𝑀‘𝑇)𝐼𝑇))
56	1, 2, 3, 20, 29, 31, 32, 42, 55	btwnlng2 28944	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝑇) ∈ (𝑉𝐿𝑇))
57	1, 2, 3, 20, 29, 31, 42, 32, 44, 56	tglineelsb2 28956	. . . . . . . . 9 ⊢ (𝜑 → (𝑉𝐿𝑇) = (𝑉𝐿(𝑀‘𝑇)))
58	57	difeq2d 4081	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∖ (𝑉𝐿𝑇)) = (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))))
59	58	eleq2d 2851	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ↔ 𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))))
60	58	eleq2d 2851	. . . . . . 7 ⊢ (𝜑 → (𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ↔ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))))
61	59, 60	anbi12d 644	. . . . . 6 ⊢ (𝜑 → ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ↔ (𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))))))
62	57	rexeqdv 3326	. . . . . 6 ⊢ (𝜑 → (∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑) ↔ ∃𝑡 ∈ (𝑉𝐿(𝑀‘𝑇))𝑡 ∈ (𝑐𝐼𝑑)))
63	61, 62	anbi12d 644	. . . . 5 ⊢ (𝜑 → (((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑)) ↔ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))) ∧ ∃𝑡 ∈ (𝑉𝐿(𝑀‘𝑇))𝑡 ∈ (𝑐𝐼𝑑))))
64	63	opabbidv 5179	. . . 4 ⊢ (𝜑 → {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑))} = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))) ∧ ∃𝑡 ∈ (𝑉𝐿(𝑀‘𝑇))𝑡 ∈ (𝑐𝐼𝑑))})
65		tgaaddcpbl.q	. . . 4 ⊢ 𝑄 = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑))}
66	64, 65, 19	3eqtr4g 2825	. . 3 ⊢ (𝜑 → 𝑄 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ∧ 𝑏 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))) ∧ ∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑎𝐼𝑏))})
67		tgaaddcpbl.5	. . 3 ⊢ (𝜑 → 𝑈𝑄𝑊)
68	66, 67	breqdi 5126	. 2 ⊢ (𝜑 → 𝑈{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ∧ 𝑏 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))) ∧ ∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑎𝐼𝑏))}𝑊)
69	4	a1i 11	. . . 4 ⊢ (𝜑 → ∼ = (cgrA‘𝐺))
70	69	eqcomd 2771	. . 3 ⊢ (𝜑 → (cgrA‘𝐺) = ∼ )
71		tgaaddcpbl.6	. . . . . . 7 ⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)
72	69, 71	breqdi 5126	. . . . . 6 ⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑇”⟩)
73	1, 2, 27, 20, 35, 21, 22, 33, 29, 31, 72	cgraswaplr 29187	. . . . 5 ⊢ (𝜑 → ⟨“𝑆𝑌𝑋”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑈”⟩)
74		tgaaddcpbllem2.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑆𝐼𝑅))
75	1, 27, 2, 20, 32, 29, 31, 55	tgbtwncom 28808	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ (𝑇𝐼(𝑀‘𝑇)))
76	1, 2, 27, 20, 22, 21, 35, 31, 29, 33, 26, 32, 73, 74, 75, 41, 45	sacgr 29193	. . . 4 ⊢ (𝜑 → ⟨“𝑅𝑌𝑋”⟩(cgrA‘𝐺)⟨“(𝑀‘𝑇)𝑉𝑈”⟩)
77	1, 2, 27, 20, 26, 21, 35, 32, 29, 33, 76	cgraswaplr 29187	. . 3 ⊢ (𝜑 → ⟨“𝑋𝑌𝑅”⟩(cgrA‘𝐺)⟨“𝑈𝑉(𝑀‘𝑇)”⟩)
78	70, 77	breqdi 5126	. 2 ⊢ (𝜑 → ⟨“𝑋𝑌𝑅”⟩ ∼ ⟨“𝑈𝑉(𝑀‘𝑇)”⟩)
79		tgaaddcpbl.7	. . . . 5 ⊢ (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)
80	69, 79	breqdi 5126	. . . 4 ⊢ (𝜑 → ⟨“𝑆𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑊”⟩)
81	1, 2, 27, 20, 22, 21, 36, 31, 29, 34, 26, 32, 80, 74, 75, 41, 45	sacgr 29193	. . 3 ⊢ (𝜑 → ⟨“𝑅𝑌𝑍”⟩(cgrA‘𝐺)⟨“(𝑀‘𝑇)𝑉𝑊”⟩)
82	70, 81	breqdi 5126	. 2 ⊢ (𝜑 → ⟨“𝑅𝑌𝑍”⟩ ∼ ⟨“(𝑀‘𝑇)𝑉𝑊”⟩)
83		tgaaddcpbllem2.k	. 2 ⊢ 𝐾 = (hlG‘𝐺)
84	1, 2, 83, 26, 35, 21, 20, 40	hlid 28932	. 2 ⊢ (𝜑 → 𝑅(𝐾‘𝑌)𝑅)
85	1, 2, 3, 4, 8, 19, 20, 26, 32, 33, 29, 34, 35, 21, 36, 41, 45, 54, 68, 78, 82, 38, 83, 46, 37, 84	tgaaddcpbllem1 29203	1 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091   ∖ cdif 3903   class class class wbr 5111  {copab 5175  ‘cfv 6540  (class class class)co 7419  ⟨“cs3 14903  Basecbs 17291  distcds 17341  TarskiGcstrkg 28747  Itvcitv 28753  LineGclng 28754  hlGchlg 28920  pInvGcmir 28980  cgrAccgra 29169

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-er 8700  df-map 8832  df-pm 8833  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-dju 9903  df-card 9941  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-3 12319  df-n0 12520  df-xnn0 12593  df-z 12607  df-uz 12879  df-fz 13552  df-fzo 13700  df-hash 14385  df-word 14569  df-concat 14626  df-s1 14653  df-s2 14909  df-s3 14910  df-trkgc 28768  df-trkgb 28769  df-trkgcb 28770  df-trkgld 28772  df-trkg 28773  df-cgrg 28831  df-leg 28903  df-hlg 28921  df-mir 28981  df-rag 29025  df-perpg 29027  df-hpg 29091  df-mid 29134  df-lmi 29135  df-cgra 29170

This theorem is used by:  tgaaddcpbllem3  29205