Theorem tgaaddcpbllem2 29229

Description: Lemma for tgaaddcpbl 29231. (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 2843	. . . . 5 ⊢ (𝑒 = 𝑠 → (𝑒 ∈ (𝑎𝐼𝑏) ↔ 𝑠 ∈ (𝑎𝐼𝑏)))
6	5	cbvrexvw 3241	. . . 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 2843	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ↔ 𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))))
10		eleq1w 2843	. . . . 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 7422	. . . . . . 7 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑎𝐼𝑏) = (𝑐𝐼𝑑))
13	12	eleq2d 2846	. . . . . 6 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑓 ∈ (𝑎𝐼𝑏) ↔ 𝑓 ∈ (𝑐𝐼𝑑)))
14	13	rexbidv 3186	. . . . 5 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑎𝐼𝑏) ↔ ∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑐𝐼𝑑)))
15		eleq1w 2843	. . . . . 6 ⊢ (𝑓 = 𝑡 → (𝑓 ∈ (𝑐𝐼𝑑) ↔ 𝑡 ∈ (𝑐𝐼𝑑)))
16	15	cbvrexvw 3241	. . . . 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 28977	. . 3 ⊢ (𝜑 → (𝑌𝐿𝑆) ∈ ran 𝐿)
25		tgaaddcpbllem2.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐿𝑆))
26	1, 3, 2, 20, 24, 25	tglnpt 28891	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
27		eqid 2760	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
28		eqid 2760	. . 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 29012	. 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 3055	. . . 4 ⊢ (𝜑 → ¬ 𝑅 = 𝑌)
40	39	neqned 2962	. . 3 ⊢ (𝜑 → 𝑅 ≠ 𝑌)
41	40	necomd 3010	. 2 ⊢ (𝜑 → 𝑌 ≠ 𝑅)
42		tgaaddcpbl.3	. . . . 5 ⊢ (𝜑 → 𝑉 ≠ 𝑇)
43	42	necomd 3010	. . . 4 ⊢ (𝜑 → 𝑇 ≠ 𝑉)
44	1, 27, 2, 3, 28, 20, 29, 30, 31, 43	mirne 29018	. . 3 ⊢ (𝜑 → (𝑀‘𝑇) ≠ 𝑉)
45	44	necomd 3010	. 2 ⊢ (𝜑 → 𝑉 ≠ (𝑀‘𝑇))
46	1, 2, 3, 20, 21, 26, 41	tglinerflx2 28981	. . 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 29096	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
50	1, 2, 3, 20, 21, 22, 23, 26, 40, 25	tglineelsb2 28979	. . . 4 ⊢ (𝜑 → (𝑌𝐿𝑆) = (𝑌𝐿𝑅))
51	49, 50	neleqtrd 2882	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑅))
52	1, 27, 2, 47, 3, 24, 20, 35, 36, 48	oppne2 29097	. . . 4 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑆))
53	52, 50	neleqtrd 2882	. . 3 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑅))
54	1, 27, 2, 8, 35, 36, 46, 51, 53, 37	islnoppd 29095	. 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 29009	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ ((𝑀‘𝑇)𝐼𝑇))
56	1, 2, 3, 20, 29, 31, 32, 42, 55	btwnlng2 28967	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝑇) ∈ (𝑉𝐿𝑇))
57	1, 2, 3, 20, 29, 31, 42, 32, 44, 56	tglineelsb2 28979	. . . . . . . . 9 ⊢ (𝜑 → (𝑉𝐿𝑇) = (𝑉𝐿(𝑀‘𝑇)))
58	57	difeq2d 4074	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∖ (𝑉𝐿𝑇)) = (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))))
59	58	eleq2d 2846	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ↔ 𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))))
60	58	eleq2d 2846	. . . . . . 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 3320	. . . . . 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 2820	. . 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 2766	. . 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 29212	. . . . 5 ⊢ (𝜑 → ⟨“𝑆𝑌𝑋”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑈”⟩)
74		tgaaddcpbllem2.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑆𝐼𝑅))
75	1, 27, 2, 20, 32, 29, 31, 55	tgbtwncom 28830	. . . . 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 29218	. . . 4 ⊢ (𝜑 → ⟨“𝑅𝑌𝑋”⟩(cgrA‘𝐺)⟨“(𝑀‘𝑇)𝑉𝑈”⟩)
77	1, 2, 27, 20, 26, 21, 35, 32, 29, 33, 76	cgraswaplr 29212	. . 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 29218	. . 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 28954	. 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 29228	1 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086   ∖ cdif 3896   class class class wbr 5103  {copab 5167  ‘cfv 6533  (class class class)co 7413  ⟨“cs3 14913  Basecbs 17301  distcds 17351  TarskiGcstrkg 28768  Itvcitv 28774  LineGclng 28775  hlGchlg 28942  pInvGcmir 29003  cgrAccgra 29193

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

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 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 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 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 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-er 8696  df-map 8828  df-pm 8829  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-dju 9906  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-n0 12529  df-xnn0 12602  df-z 12616  df-uz 12888  df-fz 13562  df-fzo 13710  df-hash 14395  df-word 14579  df-concat 14636  df-s1 14663  df-s2 14919  df-s3 14920  df-trkgc 28789  df-trkgb 28790  df-trkgcb 28791  df-trkgld 28793  df-trkg 28794  df-cgrg 28853  df-leg 28925  df-hlg 28943  df-mir 29004  df-rag 29048  df-perpg 29050  df-hpg 29115  df-mid 29158  df-lmi 29159  df-cgra 29194

This theorem is used by:  tgaaddcpbllem3  29230