Theorem tgaaddcpbllem2 29343

Description: Lemma for tgaaddcpbl 29345. (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 7427	. . . . . . 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 29091	. . 3 ⊢ (𝜑 → (𝑌𝐿𝑆) ∈ ran 𝐿)
25		tgaaddcpbllem2.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐿𝑆))
26	1, 3, 2, 20, 24, 25	tglnpt 29005	. 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 29126	. 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 29132	. . 3 ⊢ (𝜑 → (𝑀‘𝑇) ≠ 𝑉)
45	44	necomd 3011	. 2 ⊢ (𝜑 → 𝑉 ≠ (𝑀‘𝑇))
46	1, 2, 3, 20, 21, 26, 41	tglinerflx2 29095	. . 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 29210	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
50	1, 2, 3, 20, 21, 22, 23, 26, 40, 25	tglineelsb2 29093	. . . 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 29211	. . . 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 29209	. 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 29123	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ ((𝑀‘𝑇)𝐼𝑇))
56	1, 2, 3, 20, 29, 31, 32, 42, 55	btwnlng2 29081	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀‘𝑇) ∈ (𝑉𝐿𝑇))
57	1, 2, 3, 20, 29, 31, 42, 32, 44, 56	tglineelsb2 29093	. . . . . . . . 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 29326	. . . . 5 ⊢ (𝜑 → ⟨“𝑆𝑌𝑋”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑈”⟩)
74		tgaaddcpbllem2.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑆𝐼𝑅))
75	1, 27, 2, 20, 32, 29, 31, 55	tgbtwncom 28944	. . . . 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 29332	. . . 4 ⊢ (𝜑 → ⟨“𝑅𝑌𝑋”⟩(cgrA‘𝐺)⟨“(𝑀‘𝑇)𝑉𝑈”⟩)
77	1, 2, 27, 20, 26, 21, 35, 32, 29, 33, 76	cgraswaplr 29326	. . 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 29332	. . 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 29068	. 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 29342	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 6537  (class class class)co 7418  ⟨“cs3 14986  Basecbs 17380  distcds 17430  TarskiGcstrkg 28882  Itvcitv 28888  LineGclng 28889  hlGchlg 29056  pInvGcmir 29117  cgrAccgra 29307

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 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

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 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-oadd 8473  df-er 8710  df-map 8842  df-pm 8843  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-dju 9975  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-n0 12600  df-xnn0 12673  df-z 12687  df-uz 12959  df-fz 13633  df-fzo 13782  df-hash 14468  df-word 14652  df-concat 14709  df-s1 14736  df-s2 14992  df-s3 14993  df-trkgc 28903  df-trkgb 28904  df-trkgcb 28905  df-trkgld 28907  df-trkg 28908  df-cgrg 28967  df-leg 29039  df-hlg 29057  df-mir 29118  df-rag 29162  df-perpg 29164  df-hpg 29229  df-mid 29272  df-lmi 29273  df-cgra 29308

This theorem is used by:  tgaaddcpbllem3  29344