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Theorem isubgr3stgrlem4 47936
Description: Lemma 4 for isubgr3stgr 47942. (Contributed by AV, 24-Sep-2025.)
Hypotheses
Ref Expression
isubgr3stgr.v 𝑉 = (Vtx‘𝐺)
isubgr3stgr.u 𝑈 = (𝐺 NeighbVtx 𝑋)
isubgr3stgr.c 𝐶 = (𝐺 ClNeighbVtx 𝑋)
isubgr3stgr.n 𝑁 ∈ ℕ0
isubgr3stgr.s 𝑆 = (StarGr‘𝑁)
isubgr3stgr.w 𝑊 = (Vtx‘𝑆)
isubgr3stgr.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
isubgr3stgrlem4 ((𝐴 = 𝑋 ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝐴, 𝐵}) = {0, 𝑧})
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝐶   𝑧,𝐹   𝑧,𝑁   𝑧,𝑊   𝑧,𝑋
Allowed substitution hints:   𝑆(𝑧)   𝑈(𝑧)   𝐸(𝑧)   𝐺(𝑧)   𝑉(𝑧)

Proof of Theorem isubgr3stgrlem4
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq2 4734 . . . . . 6 (𝑧 = (𝐹𝐵) → {0, 𝑧} = {0, (𝐹𝐵)})
21eqeq2d 2748 . . . . 5 (𝑧 = (𝐹𝐵) → ((𝐹 “ {𝑋, 𝐵}) = {0, 𝑧} ↔ (𝐹 “ {𝑋, 𝐵}) = {0, (𝐹𝐵)}))
3 f1of 6848 . . . . . . . . 9 (𝐹:𝐶1-1-onto𝑊𝐹:𝐶𝑊)
43adantr 480 . . . . . . . 8 ((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) → 𝐹:𝐶𝑊)
54adantr 480 . . . . . . 7 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → 𝐹:𝐶𝑊)
6 simpr3 1197 . . . . . . 7 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → 𝐵𝐶)
75, 6ffvelcdmd 7105 . . . . . 6 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → (𝐹𝐵) ∈ 𝑊)
8 isubgr3stgr.w . . . . . . . . . 10 𝑊 = (Vtx‘𝑆)
9 isubgr3stgr.s . . . . . . . . . . 11 𝑆 = (StarGr‘𝑁)
109fveq2i 6909 . . . . . . . . . 10 (Vtx‘𝑆) = (Vtx‘(StarGr‘𝑁))
11 isubgr3stgr.n . . . . . . . . . . 11 𝑁 ∈ ℕ0
12 stgrvtx 47921 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
1311, 12ax-mp 5 . . . . . . . . . 10 (Vtx‘(StarGr‘𝑁)) = (0...𝑁)
148, 10, 133eqtri 2769 . . . . . . . . 9 𝑊 = (0...𝑁)
1514eleq2i 2833 . . . . . . . 8 ((𝐹𝐵) ∈ 𝑊 ↔ (𝐹𝐵) ∈ (0...𝑁))
16 fz0sn0fz1 13685 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (0...𝑁) = ({0} ∪ (1...𝑁)))
1711, 16ax-mp 5 . . . . . . . . 9 (0...𝑁) = ({0} ∪ (1...𝑁))
1817eleq2i 2833 . . . . . . . 8 ((𝐹𝐵) ∈ (0...𝑁) ↔ (𝐹𝐵) ∈ ({0} ∪ (1...𝑁)))
19 elun 4153 . . . . . . . . 9 ((𝐹𝐵) ∈ ({0} ∪ (1...𝑁)) ↔ ((𝐹𝐵) ∈ {0} ∨ (𝐹𝐵) ∈ (1...𝑁)))
20 fvex 6919 . . . . . . . . . . 11 (𝐹𝐵) ∈ V
2120elsn 4641 . . . . . . . . . 10 ((𝐹𝐵) ∈ {0} ↔ (𝐹𝐵) = 0)
2221orbi1i 914 . . . . . . . . 9 (((𝐹𝐵) ∈ {0} ∨ (𝐹𝐵) ∈ (1...𝑁)) ↔ ((𝐹𝐵) = 0 ∨ (𝐹𝐵) ∈ (1...𝑁)))
2319, 22bitri 275 . . . . . . . 8 ((𝐹𝐵) ∈ ({0} ∪ (1...𝑁)) ↔ ((𝐹𝐵) = 0 ∨ (𝐹𝐵) ∈ (1...𝑁)))
2415, 18, 233bitri 297 . . . . . . 7 ((𝐹𝐵) ∈ 𝑊 ↔ ((𝐹𝐵) = 0 ∨ (𝐹𝐵) ∈ (1...𝑁)))
25 eqeq2 2749 . . . . . . . . . . 11 ((𝐹𝑋) = 0 → ((𝐹𝐵) = (𝐹𝑋) ↔ (𝐹𝐵) = 0))
2625adantl 481 . . . . . . . . . 10 ((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) → ((𝐹𝐵) = (𝐹𝑋) ↔ (𝐹𝐵) = 0))
2726adantr 480 . . . . . . . . 9 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → ((𝐹𝐵) = (𝐹𝑋) ↔ (𝐹𝐵) = 0))
28 f1of1 6847 . . . . . . . . . . . 12 (𝐹:𝐶1-1-onto𝑊𝐹:𝐶1-1𝑊)
29 dff14a 7290 . . . . . . . . . . . . 13 (𝐹:𝐶1-1𝑊 ↔ (𝐹:𝐶𝑊 ∧ ∀𝑎𝐶𝑏𝐶 (𝑎𝑏 → (𝐹𝑎) ≠ (𝐹𝑏))))
30 simpl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝑋𝑏 = 𝐵) → 𝑎 = 𝑋)
31 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝑋𝑏 = 𝐵) → 𝑏 = 𝐵)
3230, 31neeq12d 3002 . . . . . . . . . . . . . . . . . 18 ((𝑎 = 𝑋𝑏 = 𝐵) → (𝑎𝑏𝑋𝐵))
33 fveq2 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑋 → (𝐹𝑎) = (𝐹𝑋))
3433adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝑋𝑏 = 𝐵) → (𝐹𝑎) = (𝐹𝑋))
35 fveq2 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝐵 → (𝐹𝑏) = (𝐹𝐵))
3635adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑎 = 𝑋𝑏 = 𝐵) → (𝐹𝑏) = (𝐹𝐵))
3734, 36neeq12d 3002 . . . . . . . . . . . . . . . . . 18 ((𝑎 = 𝑋𝑏 = 𝐵) → ((𝐹𝑎) ≠ (𝐹𝑏) ↔ (𝐹𝑋) ≠ (𝐹𝐵)))
3832, 37imbi12d 344 . . . . . . . . . . . . . . . . 17 ((𝑎 = 𝑋𝑏 = 𝐵) → ((𝑎𝑏 → (𝐹𝑎) ≠ (𝐹𝑏)) ↔ (𝑋𝐵 → (𝐹𝑋) ≠ (𝐹𝐵))))
3938rspc2gv 3632 . . . . . . . . . . . . . . . 16 ((𝑋𝐶𝐵𝐶) → (∀𝑎𝐶𝑏𝐶 (𝑎𝑏 → (𝐹𝑎) ≠ (𝐹𝑏)) → (𝑋𝐵 → (𝐹𝑋) ≠ (𝐹𝐵))))
40393adant1 1131 . . . . . . . . . . . . . . 15 ((𝑋𝐵𝑋𝐶𝐵𝐶) → (∀𝑎𝐶𝑏𝐶 (𝑎𝑏 → (𝐹𝑎) ≠ (𝐹𝑏)) → (𝑋𝐵 → (𝐹𝑋) ≠ (𝐹𝐵))))
41 id 22 . . . . . . . . . . . . . . . . 17 ((𝑋𝐵 → (𝐹𝑋) ≠ (𝐹𝐵)) → (𝑋𝐵 → (𝐹𝑋) ≠ (𝐹𝐵)))
42 eqneqall 2951 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑋) = (𝐹𝐵) → ((𝐹𝑋) ≠ (𝐹𝐵) → (𝐹𝐵) ∈ (1...𝑁)))
4342eqcoms 2745 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐵) = (𝐹𝑋) → ((𝐹𝑋) ≠ (𝐹𝐵) → (𝐹𝐵) ∈ (1...𝑁)))
4443com12 32 . . . . . . . . . . . . . . . . 17 ((𝐹𝑋) ≠ (𝐹𝐵) → ((𝐹𝐵) = (𝐹𝑋) → (𝐹𝐵) ∈ (1...𝑁)))
4541, 44syl6com 37 . . . . . . . . . . . . . . . 16 (𝑋𝐵 → ((𝑋𝐵 → (𝐹𝑋) ≠ (𝐹𝐵)) → ((𝐹𝐵) = (𝐹𝑋) → (𝐹𝐵) ∈ (1...𝑁))))
46453ad2ant1 1134 . . . . . . . . . . . . . . 15 ((𝑋𝐵𝑋𝐶𝐵𝐶) → ((𝑋𝐵 → (𝐹𝑋) ≠ (𝐹𝐵)) → ((𝐹𝐵) = (𝐹𝑋) → (𝐹𝐵) ∈ (1...𝑁))))
4740, 46syld 47 . . . . . . . . . . . . . 14 ((𝑋𝐵𝑋𝐶𝐵𝐶) → (∀𝑎𝐶𝑏𝐶 (𝑎𝑏 → (𝐹𝑎) ≠ (𝐹𝑏)) → ((𝐹𝐵) = (𝐹𝑋) → (𝐹𝐵) ∈ (1...𝑁))))
4847adantld 490 . . . . . . . . . . . . 13 ((𝑋𝐵𝑋𝐶𝐵𝐶) → ((𝐹:𝐶𝑊 ∧ ∀𝑎𝐶𝑏𝐶 (𝑎𝑏 → (𝐹𝑎) ≠ (𝐹𝑏))) → ((𝐹𝐵) = (𝐹𝑋) → (𝐹𝐵) ∈ (1...𝑁))))
4929, 48biimtrid 242 . . . . . . . . . . . 12 ((𝑋𝐵𝑋𝐶𝐵𝐶) → (𝐹:𝐶1-1𝑊 → ((𝐹𝐵) = (𝐹𝑋) → (𝐹𝐵) ∈ (1...𝑁))))
5028, 49syl5com 31 . . . . . . . . . . 11 (𝐹:𝐶1-1-onto𝑊 → ((𝑋𝐵𝑋𝐶𝐵𝐶) → ((𝐹𝐵) = (𝐹𝑋) → (𝐹𝐵) ∈ (1...𝑁))))
5150adantr 480 . . . . . . . . . 10 ((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) → ((𝑋𝐵𝑋𝐶𝐵𝐶) → ((𝐹𝐵) = (𝐹𝑋) → (𝐹𝐵) ∈ (1...𝑁))))
5251imp 406 . . . . . . . . 9 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → ((𝐹𝐵) = (𝐹𝑋) → (𝐹𝐵) ∈ (1...𝑁)))
5327, 52sylbird 260 . . . . . . . 8 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → ((𝐹𝐵) = 0 → (𝐹𝐵) ∈ (1...𝑁)))
54 idd 24 . . . . . . . 8 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → ((𝐹𝐵) ∈ (1...𝑁) → (𝐹𝐵) ∈ (1...𝑁)))
5553, 54jaod 860 . . . . . . 7 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → (((𝐹𝐵) = 0 ∨ (𝐹𝐵) ∈ (1...𝑁)) → (𝐹𝐵) ∈ (1...𝑁)))
5624, 55biimtrid 242 . . . . . 6 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → ((𝐹𝐵) ∈ 𝑊 → (𝐹𝐵) ∈ (1...𝑁)))
577, 56mpd 15 . . . . 5 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → (𝐹𝐵) ∈ (1...𝑁))
58 f1ofn 6849 . . . . . . . . . 10 (𝐹:𝐶1-1-onto𝑊𝐹 Fn 𝐶)
5958adantr 480 . . . . . . . . 9 ((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) → 𝐹 Fn 𝐶)
60 3simpc 1151 . . . . . . . . 9 ((𝑋𝐵𝑋𝐶𝐵𝐶) → (𝑋𝐶𝐵𝐶))
6159, 60anim12i 613 . . . . . . . 8 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → (𝐹 Fn 𝐶 ∧ (𝑋𝐶𝐵𝐶)))
62 3anass 1095 . . . . . . . 8 ((𝐹 Fn 𝐶𝑋𝐶𝐵𝐶) ↔ (𝐹 Fn 𝐶 ∧ (𝑋𝐶𝐵𝐶)))
6361, 62sylibr 234 . . . . . . 7 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → (𝐹 Fn 𝐶𝑋𝐶𝐵𝐶))
64 fnimapr 6992 . . . . . . 7 ((𝐹 Fn 𝐶𝑋𝐶𝐵𝐶) → (𝐹 “ {𝑋, 𝐵}) = {(𝐹𝑋), (𝐹𝐵)})
6563, 64syl 17 . . . . . 6 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → (𝐹 “ {𝑋, 𝐵}) = {(𝐹𝑋), (𝐹𝐵)})
66 simpr 484 . . . . . . . 8 ((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) → (𝐹𝑋) = 0)
6766adantr 480 . . . . . . 7 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → (𝐹𝑋) = 0)
6867preq1d 4739 . . . . . 6 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → {(𝐹𝑋), (𝐹𝐵)} = {0, (𝐹𝐵)})
6965, 68eqtrd 2777 . . . . 5 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → (𝐹 “ {𝑋, 𝐵}) = {0, (𝐹𝐵)})
702, 57, 69rspcedvdw 3625 . . . 4 (((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝑋𝐵𝑋𝐶𝐵𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑋, 𝐵}) = {0, 𝑧})
7170ex 412 . . 3 ((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) → ((𝑋𝐵𝑋𝐶𝐵𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑋, 𝐵}) = {0, 𝑧}))
72 neeq1 3003 . . . . 5 (𝐴 = 𝑋 → (𝐴𝐵𝑋𝐵))
73 eleq1 2829 . . . . 5 (𝐴 = 𝑋 → (𝐴𝐶𝑋𝐶))
7472, 733anbi12d 1439 . . . 4 (𝐴 = 𝑋 → ((𝐴𝐵𝐴𝐶𝐵𝐶) ↔ (𝑋𝐵𝑋𝐶𝐵𝐶)))
75 preq1 4733 . . . . . . 7 (𝐴 = 𝑋 → {𝐴, 𝐵} = {𝑋, 𝐵})
7675imaeq2d 6078 . . . . . 6 (𝐴 = 𝑋 → (𝐹 “ {𝐴, 𝐵}) = (𝐹 “ {𝑋, 𝐵}))
7776eqeq1d 2739 . . . . 5 (𝐴 = 𝑋 → ((𝐹 “ {𝐴, 𝐵}) = {0, 𝑧} ↔ (𝐹 “ {𝑋, 𝐵}) = {0, 𝑧}))
7877rexbidv 3179 . . . 4 (𝐴 = 𝑋 → (∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝐴, 𝐵}) = {0, 𝑧} ↔ ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑋, 𝐵}) = {0, 𝑧}))
7974, 78imbi12d 344 . . 3 (𝐴 = 𝑋 → (((𝐴𝐵𝐴𝐶𝐵𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝐴, 𝐵}) = {0, 𝑧}) ↔ ((𝑋𝐵𝑋𝐶𝐵𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝑋, 𝐵}) = {0, 𝑧})))
8071, 79imbitrrid 246 . 2 (𝐴 = 𝑋 → ((𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝐴, 𝐵}) = {0, 𝑧})))
81803imp 1111 1 ((𝐴 = 𝑋 ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝐴, 𝐵}) = {0, 𝑧})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cun 3949  {csn 4626  {cpr 4628  cima 5688   Fn wfn 6556  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  0cn0 12526  ...cfz 13547  Vtxcvtx 29013  Edgcedg 29064   NeighbVtx cnbgr 29349   ClNeighbVtx cclnbgr 47805  StarGrcstgr 47918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-hash 14370  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-edgf 29004  df-vtx 29015  df-stgr 47919
This theorem is referenced by:  isubgr3stgrlem6  47938
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