Theorem tgaaddcpbllem2 29350

Description: Lemma for tgaaddcpbl 29352. (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 2844	. . . . 5 ⊢ (𝑒 = 𝑠 → (𝑒 ∈ (𝑎𝐼𝑏) ↔ 𝑠 ∈ (𝑎𝐼𝑏)))
6	5	cbvrexvw 3242	. . . 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 5172	. 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 2844	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ↔ 𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))))
10		eleq1w 2844	. . . . 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 7429	. . . . . . 7 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑎𝐼𝑏) = (𝑐𝐼𝑑))
13	12	eleq2d 2847	. . . . . 6 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑓 ∈ (𝑎𝐼𝑏) ↔ 𝑓 ∈ (𝑐𝐼𝑑)))
14	13	rexbidv 3187	. . . . 5 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑎𝐼𝑏) ↔ ∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑐𝐼𝑑)))
15		eleq1w 2844	. . . . . 6 ⊢ (𝑓 = 𝑡 → (𝑓 ∈ (𝑐𝐼𝑑) ↔ 𝑡 ∈ (𝑐𝐼𝑑)))
16	15	cbvrexvw 3242	. . . . 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 5178	. 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 29098	. . 3 ⊢ (𝜑 → (𝑌𝐿𝑆) ∈ ran 𝐿)
25		tgaaddcpbllem2.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐿𝑆))
26	1, 3, 2, 20, 24, 25	tglnpt 29012	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
27		eqid 2761	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
28		eqid 2761	. . 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 29133	. 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 3056	. . . 4 ⊢ (𝜑 → ¬ 𝑅 = 𝑌)
40	39	neqned 2963	. . 3 ⊢ (𝜑 → 𝑅 ≠ 𝑌)
41	40	necomd 3011	. 2 ⊢ (𝜑 → 𝑌 ≠ 𝑅)
42		tgaaddcpbl.3	. . . . 5 ⊢ (𝜑 → 𝑉 ≠ 𝑇)
43	42	necomd 3011	. . . 4 ⊢ (𝜑 → 𝑇 ≠ 𝑉)
44	1, 27, 2, 3, 28, 20, 29, 30, 31, 43	mirne 29139	. . 3 ⊢ (𝜑 → (𝑀‘𝑇) ≠ 𝑉)
45	44	necomd 3011	. 2 ⊢ (𝜑 → 𝑉 ≠ (𝑀‘𝑇))
46	1, 2, 3, 20, 21, 26, 41	tglinerflx2 29102	. . 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 29217	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
50	1, 2, 3, 20, 21, 22, 23, 26, 40, 25	tglineelsb2 29100	. . . 4 ⊢ (𝜑 → (𝑌𝐿𝑆) = (𝑌𝐿𝑅))
51	49, 50	neleqtrd 2883	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑅))
52	1, 27, 2, 47, 3, 24, 20, 35, 36, 48	oppne2 29218	. . . 4 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑆))
53	52, 50	neleqtrd 2883	. . 3 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑅))
54	1, 27, 2, 8, 35, 36, 46, 51, 53, 37	islnoppd 29216	. 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 29130	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ ((𝑀‘𝑇)𝐼𝑇))
56	1, 2, 3, 20, 29, 31, 32, 42, 55	btwnlng2 29088	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝑇) ∈ (𝑉𝐿𝑇))
57	1, 2, 3, 20, 29, 31, 42, 32, 44, 56	tglineelsb2 29100	. . . . . . . . 9 ⊢ (𝜑 → (𝑉𝐿𝑇) = (𝑉𝐿(𝑀‘𝑇)))
58	57	difeq2d 4074	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∖ (𝑉𝐿𝑇)) = (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))))
59	58	eleq2d 2847	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ↔ 𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))))
60	58	eleq2d 2847	. . . . . . 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 3321	. . . . . 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 5171	. . . 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 2821	. . 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 5118	. 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 2767	. . 3 ⊢ (𝜑 → (cgrA‘𝐺) = ∼ )
71		tgaaddcpbl.6	. . . . . . 7 ⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)
72	69, 71	breqdi 5118	. . . . . 6 ⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑇”⟩)
73	1, 2, 27, 20, 35, 21, 22, 33, 29, 31, 72	cgraswaplr 29333	. . . . 5 ⊢ (𝜑 → ⟨“𝑆𝑌𝑋”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑈”⟩)
74		tgaaddcpbllem2.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑆𝐼𝑅))
75	1, 27, 2, 20, 32, 29, 31, 55	tgbtwncom 28951	. . . . 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 29339	. . . 4 ⊢ (𝜑 → ⟨“𝑅𝑌𝑋”⟩(cgrA‘𝐺)⟨“(𝑀‘𝑇)𝑉𝑈”⟩)
77	1, 2, 27, 20, 26, 21, 35, 32, 29, 33, 76	cgraswaplr 29333	. . 3 ⊢ (𝜑 → ⟨“𝑋𝑌𝑅”⟩(cgrA‘𝐺)⟨“𝑈𝑉(𝑀‘𝑇)”⟩)
78	70, 77	breqdi 5118	. 2 ⊢ (𝜑 → ⟨“𝑋𝑌𝑅”⟩ ∼ ⟨“𝑈𝑉(𝑀‘𝑇)”⟩)
79		tgaaddcpbl.7	. . . . 5 ⊢ (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)
80	69, 79	breqdi 5118	. . . 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 29339	. . 3 ⊢ (𝜑 → ⟨“𝑅𝑌𝑍”⟩(cgrA‘𝐺)⟨“(𝑀‘𝑇)𝑉𝑊”⟩)
82	70, 81	breqdi 5118	. 2 ⊢ (𝜑 → ⟨“𝑅𝑌𝑍”⟩ ∼ ⟨“(𝑀‘𝑇)𝑉𝑊”⟩)
83		tgaaddcpbllem2.k	. 2 ⊢ 𝐾 = (hlG‘𝐺)
84	1, 2, 83, 26, 35, 21, 20, 40	hlid 29075	. 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 29349	1 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087   ∖ cdif 3896   class class class wbr 5103  {copab 5167  ‘cfv 6538  (class class class)co 7420  ⟨“cs3 14993  Basecbs 17387  distcds 17437  TarskiGcstrkg 28889  Itvcitv 28895  LineGclng 28896  hlGchlg 29063  pInvGcmir 29124  cgrAccgra 29314

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  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 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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-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-trkgc 28910  df-trkgb 28911  df-trkgcb 28912  df-trkgld 28914  df-trkg 28915  df-cgrg 28974  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

This theorem is used by:  tgaaddcpbllem3  29351