Theorem tgaaddcpbllem2 29407

Description: Lemma for tgaaddcpbl 29409. (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 2847	. . . . 5 ⊢ (𝑒 = 𝑠 → (𝑒 ∈ (𝑎𝐼𝑏) ↔ 𝑠 ∈ (𝑎𝐼𝑏)))
6	5	cbvrexvw 3245	. . . 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 5178	. 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 2847	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))) ↔ 𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))))
10		eleq1w 2847	. . . . 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 7460	. . . . . . 7 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑎𝐼𝑏) = (𝑐𝐼𝑑))
13	12	eleq2d 2850	. . . . . 6 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (𝑓 ∈ (𝑎𝐼𝑏) ↔ 𝑓 ∈ (𝑐𝐼𝑑)))
14	13	rexbidv 3190	. . . . 5 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → (∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑎𝐼𝑏) ↔ ∃𝑓 ∈ (𝑉𝐿(𝑀‘𝑇))𝑓 ∈ (𝑐𝐼𝑑)))
15		eleq1w 2847	. . . . . 6 ⊢ (𝑓 = 𝑡 → (𝑓 ∈ (𝑐𝐼𝑑) ↔ 𝑡 ∈ (𝑐𝐼𝑑)))
16	15	cbvrexvw 3245	. . . . 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 5184	. 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 29155	. . 3 ⊢ (𝜑 → (𝑌𝐿𝑆) ∈ ran 𝐿)
25		tgaaddcpbllem2.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐿𝑆))
26	1, 3, 2, 20, 24, 25	tglnpt 29069	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
27		eqid 2763	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
28		eqid 2763	. . 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 29190	. 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 3059	. . . 4 ⊢ (𝜑 → ¬ 𝑅 = 𝑌)
40	39	neqned 2966	. . 3 ⊢ (𝜑 → 𝑅 ≠ 𝑌)
41	40	necomd 3014	. 2 ⊢ (𝜑 → 𝑌 ≠ 𝑅)
42		tgaaddcpbl.3	. . . . 5 ⊢ (𝜑 → 𝑉 ≠ 𝑇)
43	42	necomd 3014	. . . 4 ⊢ (𝜑 → 𝑇 ≠ 𝑉)
44	1, 27, 2, 3, 28, 20, 29, 30, 31, 43	mirne 29196	. . 3 ⊢ (𝜑 → (𝑀‘𝑇) ≠ 𝑉)
45	44	necomd 3014	. 2 ⊢ (𝜑 → 𝑉 ≠ (𝑀‘𝑇))
46	1, 2, 3, 20, 21, 26, 41	tglinerflx2 29159	. . 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 29274	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
50	1, 2, 3, 20, 21, 22, 23, 26, 40, 25	tglineelsb2 29157	. . . 4 ⊢ (𝜑 → (𝑌𝐿𝑆) = (𝑌𝐿𝑅))
51	49, 50	neleqtrd 2886	. . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑅))
52	1, 27, 2, 47, 3, 24, 20, 35, 36, 48	oppne2 29275	. . . 4 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑆))
53	52, 50	neleqtrd 2886	. . 3 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑅))
54	1, 27, 2, 8, 35, 36, 46, 51, 53, 37	islnoppd 29273	. 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 29187	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ ((𝑀‘𝑇)𝐼𝑇))
56	1, 2, 3, 20, 29, 31, 32, 42, 55	btwnlng2 29145	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝑇) ∈ (𝑉𝐿𝑇))
57	1, 2, 3, 20, 29, 31, 42, 32, 44, 56	tglineelsb2 29157	. . . . . . . . 9 ⊢ (𝜑 → (𝑉𝐿𝑇) = (𝑉𝐿(𝑀‘𝑇)))
58	57	difeq2d 4066	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∖ (𝑉𝐿𝑇)) = (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇))))
59	58	eleq2d 2850	. . . . . . 7 ⊢ (𝜑 → (𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ↔ 𝑐 ∈ (𝑃 ∖ (𝑉𝐿(𝑀‘𝑇)))))
60	58	eleq2d 2850	. . . . . . 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 3324	. . . . . 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 5177	. . . 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 2823	. . 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 5123	. 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 2769	. . 3 ⊢ (𝜑 → (cgrA‘𝐺) = ∼ )
71		tgaaddcpbl.6	. . . . . . 7 ⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)
72	69, 71	breqdi 5123	. . . . . 6 ⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑇”⟩)
73	1, 2, 27, 20, 35, 21, 22, 33, 29, 31, 72	cgraswaplr 29390	. . . . 5 ⊢ (𝜑 → ⟨“𝑆𝑌𝑋”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑈”⟩)
74		tgaaddcpbllem2.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑆𝐼𝑅))
75	1, 27, 2, 20, 32, 29, 31, 55	tgbtwncom 29008	. . . . 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 29396	. . . 4 ⊢ (𝜑 → ⟨“𝑅𝑌𝑋”⟩(cgrA‘𝐺)⟨“(𝑀‘𝑇)𝑉𝑈”⟩)
77	1, 2, 27, 20, 26, 21, 35, 32, 29, 33, 76	cgraswaplr 29390	. . 3 ⊢ (𝜑 → ⟨“𝑋𝑌𝑅”⟩(cgrA‘𝐺)⟨“𝑈𝑉(𝑀‘𝑇)”⟩)
78	70, 77	breqdi 5123	. 2 ⊢ (𝜑 → ⟨“𝑋𝑌𝑅”⟩ ∼ ⟨“𝑈𝑉(𝑀‘𝑇)”⟩)
79		tgaaddcpbl.7	. . . . 5 ⊢ (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)
80	69, 79	breqdi 5123	. . . 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 29396	. . 3 ⊢ (𝜑 → ⟨“𝑅𝑌𝑍”⟩(cgrA‘𝐺)⟨“(𝑀‘𝑇)𝑉𝑊”⟩)
82	70, 81	breqdi 5123	. 2 ⊢ (𝜑 → ⟨“𝑅𝑌𝑍”⟩ ∼ ⟨“(𝑀‘𝑇)𝑉𝑊”⟩)
83		tgaaddcpbllem2.k	. 2 ⊢ 𝐾 = (hlG‘𝐺)
84	1, 2, 83, 26, 35, 21, 20, 40	hlid 29132	. 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 29406	1 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090   ∖ cdif 4048   class class class wbr 5108  {copab 5173  ‘cfv 6566  (class class class)co 7451  ⟨“cs3 15035  Basecbs 17431  distcds 17481  TarskiGcstrkg 28946  Itvcitv 28952  LineGclng 28953  hlGchlg 29120  pInvGcmir 29181  cgrAccgra 29371

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

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 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-oadd 8518  df-er 8755  df-map 8887  df-pm 8888  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-dju 10019  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-xnn0 12720  df-z 12734  df-uz 13006  df-fz 13680  df-fzo 13829  df-hash 14516  df-word 14700  df-concat 14757  df-s1 14784  df-s2 15041  df-s3 15042  df-trkgc 28967  df-trkgb 28968  df-trkgcb 28969  df-trkgld 28971  df-trkg 28972  df-cgrg 29031  df-leg 29103  df-hlg 29121  df-mir 29182  df-rag 29226  df-perpg 29228  df-hpg 29293  df-mid 29336  df-lmi 29337  df-cgra 29372

This theorem is used by:  tgaaddcpbllem3  29408