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Theorem unitscyglem1 42177
Description: Lemma for unitscyg. (Contributed by metakunt, 13-Jul-2025.)
Hypotheses
Ref Expression
unitscyglem1.1 𝐵 = (Base‘𝐺)
unitscyglem1.2 = (.g𝐺)
unitscyglem1.3 (𝜑𝐺 ∈ Grp)
unitscyglem1.4 (𝜑𝐵 ∈ Fin)
unitscyglem1.5 (𝜑 → ∀𝑛 ∈ ℕ (♯‘{𝑥𝐵 ∣ (𝑛 𝑥) = (0g𝐺)}) ≤ 𝑛)
unitscyglem1.6 (𝜑𝐴𝐵)
Assertion
Ref Expression
unitscyglem1 (𝜑 → (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}) = ((od‘𝐺)‘𝐴))
Distinct variable groups:   ,𝑛,𝑥   𝐴,𝑛,𝑥   𝐵,𝑛,𝑥   𝑛,𝐺,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑛)

Proof of Theorem unitscyglem1
Dummy variables 𝑖 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . . . . . . 8 (𝑛 = ((od‘𝐺)‘𝐴) → (𝑛 𝑥) = (((od‘𝐺)‘𝐴) 𝑥))
21eqeq1d 2737 . . . . . . 7 (𝑛 = ((od‘𝐺)‘𝐴) → ((𝑛 𝑥) = (0g𝐺) ↔ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)))
32rabbidv 3441 . . . . . 6 (𝑛 = ((od‘𝐺)‘𝐴) → {𝑥𝐵 ∣ (𝑛 𝑥) = (0g𝐺)} = {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)})
43fveq2d 6911 . . . . 5 (𝑛 = ((od‘𝐺)‘𝐴) → (♯‘{𝑥𝐵 ∣ (𝑛 𝑥) = (0g𝐺)}) = (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}))
5 id 22 . . . . 5 (𝑛 = ((od‘𝐺)‘𝐴) → 𝑛 = ((od‘𝐺)‘𝐴))
64, 5breq12d 5161 . . . 4 (𝑛 = ((od‘𝐺)‘𝐴) → ((♯‘{𝑥𝐵 ∣ (𝑛 𝑥) = (0g𝐺)}) ≤ 𝑛 ↔ (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}) ≤ ((od‘𝐺)‘𝐴)))
7 unitscyglem1.5 . . . 4 (𝜑 → ∀𝑛 ∈ ℕ (♯‘{𝑥𝐵 ∣ (𝑛 𝑥) = (0g𝐺)}) ≤ 𝑛)
8 unitscyglem1.3 . . . . 5 (𝜑𝐺 ∈ Grp)
9 unitscyglem1.4 . . . . 5 (𝜑𝐵 ∈ Fin)
10 unitscyglem1.6 . . . . 5 (𝜑𝐴𝐵)
11 unitscyglem1.1 . . . . . 6 𝐵 = (Base‘𝐺)
12 eqid 2735 . . . . . 6 (od‘𝐺) = (od‘𝐺)
1311, 12odcl2 19598 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐵 ∈ Fin ∧ 𝐴𝐵) → ((od‘𝐺)‘𝐴) ∈ ℕ)
148, 9, 10, 13syl3anc 1370 . . . 4 (𝜑 → ((od‘𝐺)‘𝐴) ∈ ℕ)
156, 7, 14rspcdva 3623 . . 3 (𝜑 → (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}) ≤ ((od‘𝐺)‘𝐴))
16 unitscyglem1.2 . . . . . . 7 = (.g𝐺)
17 eqid 2735 . . . . . . 7 (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) = (𝑖 ∈ ℤ ↦ (𝑖 𝐴))
1811, 12, 16, 17dfod2 19597 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → ((od‘𝐺)‘𝐴) = if(ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) ∈ Fin, (♯‘ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴))), 0))
198, 10, 18syl2anc 584 . . . . 5 (𝜑 → ((od‘𝐺)‘𝐴) = if(ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) ∈ Fin, (♯‘ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴))), 0))
208adantr 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ ℤ) → 𝐺 ∈ Grp)
21 simpr 484 . . . . . . . . . 10 ((𝜑𝑖 ∈ ℤ) → 𝑖 ∈ ℤ)
2210adantr 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ ℤ) → 𝐴𝐵)
2311, 16, 20, 21, 22mulgcld 19127 . . . . . . . . 9 ((𝜑𝑖 ∈ ℤ) → (𝑖 𝐴) ∈ 𝐵)
2423fmpttd 7135 . . . . . . . 8 (𝜑 → (𝑖 ∈ ℤ ↦ (𝑖 𝐴)):ℤ⟶𝐵)
25 frn 6744 . . . . . . . 8 ((𝑖 ∈ ℤ ↦ (𝑖 𝐴)):ℤ⟶𝐵 → ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) ⊆ 𝐵)
2624, 25syl 17 . . . . . . 7 (𝜑 → ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) ⊆ 𝐵)
279, 26ssfid 9299 . . . . . 6 (𝜑 → ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) ∈ Fin)
2827iftrued 4539 . . . . 5 (𝜑 → if(ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) ∈ Fin, (♯‘ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴))), 0) = (♯‘ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴))))
2919, 28eqtrd 2775 . . . 4 (𝜑 → ((od‘𝐺)‘𝐴) = (♯‘ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴))))
30 eqid 2735 . . . . . 6 {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)} = {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}
31 fvexd 6922 . . . . . . 7 (𝜑 → (Base‘𝐺) ∈ V)
3211, 31eqeltrid 2843 . . . . . 6 (𝜑𝐵 ∈ V)
3330, 32rabexd 5346 . . . . 5 (𝜑 → {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)} ∈ V)
34 ovexd 7466 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ℤ) → (𝑖 𝐴) ∈ V)
3534fmpttd 7135 . . . . . . . . . . 11 (𝜑 → (𝑖 ∈ ℤ ↦ (𝑖 𝐴)):ℤ⟶V)
3635ffnd 6738 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) Fn ℤ)
37 fvelrnb 6969 . . . . . . . . . 10 ((𝑖 ∈ ℤ ↦ (𝑖 𝐴)) Fn ℤ → (𝑦 ∈ ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) ↔ ∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦))
3836, 37syl 17 . . . . . . . . 9 (𝜑 → (𝑦 ∈ ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) ↔ ∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦))
3938biimpa 476 . . . . . . . 8 ((𝜑𝑦 ∈ ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴))) → ∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦)
40 id 22 . . . . . . . . . . . . . 14 (((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) = 𝑦 → ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) = 𝑦)
4140eqcomd 2741 . . . . . . . . . . . . 13 (((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) = 𝑦𝑦 = ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤))
4241adantl 481 . . . . . . . . . . . 12 ((((𝜑 ∧ ∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦) ∧ 𝑤 ∈ ℤ) ∧ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) = 𝑦) → 𝑦 = ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤))
43 simpll 767 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦) ∧ 𝑤 ∈ ℤ) → 𝜑)
44 simpr 484 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦) ∧ 𝑤 ∈ ℤ) → 𝑤 ∈ ℤ)
4543, 44jca 511 . . . . . . . . . . . . . 14 (((𝜑 ∧ ∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦) ∧ 𝑤 ∈ ℤ) → (𝜑𝑤 ∈ ℤ))
46 eqidd 2736 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ ℤ) → (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) = (𝑖 ∈ ℤ ↦ (𝑖 𝐴)))
47 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ∈ ℤ) ∧ 𝑖 = 𝑤) → 𝑖 = 𝑤)
4847oveq1d 7446 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ∈ ℤ) ∧ 𝑖 = 𝑤) → (𝑖 𝐴) = (𝑤 𝐴))
49 simpr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ ℤ) → 𝑤 ∈ ℤ)
50 ovexd 7466 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ ℤ) → (𝑤 𝐴) ∈ V)
5146, 48, 49, 50fvmptd 7023 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ ℤ) → ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) = (𝑤 𝐴))
52 oveq2 7439 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑤 𝐴) → (((od‘𝐺)‘𝐴) 𝑥) = (((od‘𝐺)‘𝐴) (𝑤 𝐴)))
5352eqeq1d 2737 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑤 𝐴) → ((((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺) ↔ (((od‘𝐺)‘𝐴) (𝑤 𝐴)) = (0g𝐺)))
548adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ ℤ) → 𝐺 ∈ Grp)
5510adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ ℤ) → 𝐴𝐵)
5611, 16, 54, 49, 55mulgcld 19127 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ ℤ) → (𝑤 𝐴) ∈ 𝐵)
5714nnzd 12638 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((od‘𝐺)‘𝐴) ∈ ℤ)
5857adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ℤ) → ((od‘𝐺)‘𝐴) ∈ ℤ)
5949, 58, 553jca 1127 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ℤ) → (𝑤 ∈ ℤ ∧ ((od‘𝐺)‘𝐴) ∈ ℤ ∧ 𝐴𝐵))
6011, 16mulgass 19142 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ Grp ∧ (𝑤 ∈ ℤ ∧ ((od‘𝐺)‘𝐴) ∈ ℤ ∧ 𝐴𝐵)) → ((𝑤 · ((od‘𝐺)‘𝐴)) 𝐴) = (𝑤 (((od‘𝐺)‘𝐴) 𝐴)))
6154, 59, 60syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ ℤ) → ((𝑤 · ((od‘𝐺)‘𝐴)) 𝐴) = (𝑤 (((od‘𝐺)‘𝐴) 𝐴)))
62 eqid 2735 . . . . . . . . . . . . . . . . . . . . . 22 (0g𝐺) = (0g𝐺)
6311, 12, 16, 62odid 19571 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝐵 → (((od‘𝐺)‘𝐴) 𝐴) = (0g𝐺))
6455, 63syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ℤ) → (((od‘𝐺)‘𝐴) 𝐴) = (0g𝐺))
6564oveq2d 7447 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ℤ) → (𝑤 (((od‘𝐺)‘𝐴) 𝐴)) = (𝑤 (0g𝐺)))
6611, 16, 62mulgz 19133 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ Grp ∧ 𝑤 ∈ ℤ) → (𝑤 (0g𝐺)) = (0g𝐺))
678, 66sylan 580 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ℤ) → (𝑤 (0g𝐺)) = (0g𝐺))
6865, 67eqtrd 2775 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ ℤ) → (𝑤 (((od‘𝐺)‘𝐴) 𝐴)) = (0g𝐺))
6961, 68eqtr2d 2776 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ ℤ) → (0g𝐺) = ((𝑤 · ((od‘𝐺)‘𝐴)) 𝐴))
7059simp2d 1142 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ℤ) → ((od‘𝐺)‘𝐴) ∈ ℤ)
7170, 49, 553jca 1127 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ ℤ) → (((od‘𝐺)‘𝐴) ∈ ℤ ∧ 𝑤 ∈ ℤ ∧ 𝐴𝐵))
7211, 16mulgassr 19143 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ (((od‘𝐺)‘𝐴) ∈ ℤ ∧ 𝑤 ∈ ℤ ∧ 𝐴𝐵)) → ((𝑤 · ((od‘𝐺)‘𝐴)) 𝐴) = (((od‘𝐺)‘𝐴) (𝑤 𝐴)))
7354, 71, 72syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ ℤ) → ((𝑤 · ((od‘𝐺)‘𝐴)) 𝐴) = (((od‘𝐺)‘𝐴) (𝑤 𝐴)))
7469, 73eqtr2d 2776 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ ℤ) → (((od‘𝐺)‘𝐴) (𝑤 𝐴)) = (0g𝐺))
7553, 56, 74elrabd 3697 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ ℤ) → (𝑤 𝐴) ∈ {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)})
7651, 75eqeltrd 2839 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ℤ) → ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) ∈ {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)})
7745, 76syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ ∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦) ∧ 𝑤 ∈ ℤ) → ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) ∈ {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)})
7877adantr 480 . . . . . . . . . . . 12 ((((𝜑 ∧ ∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦) ∧ 𝑤 ∈ ℤ) ∧ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) = 𝑦) → ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) ∈ {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)})
7942, 78eqeltrd 2839 . . . . . . . . . . 11 ((((𝜑 ∧ ∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦) ∧ 𝑤 ∈ ℤ) ∧ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) = 𝑦) → 𝑦 ∈ {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)})
80 nfv 1912 . . . . . . . . . . . . . 14 𝑤((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦
81 nfv 1912 . . . . . . . . . . . . . 14 𝑧((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) = 𝑦
82 fveqeq2 6916 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → (((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦 ↔ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) = 𝑦))
8380, 81, 82cbvrexw 3305 . . . . . . . . . . . . 13 (∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦 ↔ ∃𝑤 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) = 𝑦)
8483biimpi 216 . . . . . . . . . . . 12 (∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦 → ∃𝑤 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) = 𝑦)
8584adantl 481 . . . . . . . . . . 11 ((𝜑 ∧ ∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦) → ∃𝑤 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑤) = 𝑦)
8679, 85r19.29a 3160 . . . . . . . . . 10 ((𝜑 ∧ ∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦) → 𝑦 ∈ {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)})
8786ex 412 . . . . . . . . 9 (𝜑 → (∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦𝑦 ∈ {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}))
8887adantr 480 . . . . . . . 8 ((𝜑𝑦 ∈ ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴))) → (∃𝑧 ∈ ℤ ((𝑖 ∈ ℤ ↦ (𝑖 𝐴))‘𝑧) = 𝑦𝑦 ∈ {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}))
8939, 88mpd 15 . . . . . . 7 ((𝜑𝑦 ∈ ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴))) → 𝑦 ∈ {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)})
9089ex 412 . . . . . 6 (𝜑 → (𝑦 ∈ ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) → 𝑦 ∈ {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}))
9190ssrdv 4001 . . . . 5 (𝜑 → ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) ⊆ {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)})
92 hashss 14445 . . . . 5 (({𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)} ∈ V ∧ ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴)) ⊆ {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}) → (♯‘ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴))) ≤ (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}))
9333, 91, 92syl2anc 584 . . . 4 (𝜑 → (♯‘ran (𝑖 ∈ ℤ ↦ (𝑖 𝐴))) ≤ (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}))
9429, 93eqbrtrd 5170 . . 3 (𝜑 → ((od‘𝐺)‘𝐴) ≤ (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}))
9515, 94jca 511 . 2 (𝜑 → ((♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}) ≤ ((od‘𝐺)‘𝐴) ∧ ((od‘𝐺)‘𝐴) ≤ (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)})))
96 ssrab2 4090 . . . . . . 7 {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)} ⊆ 𝐵
9796a1i 11 . . . . . 6 (𝜑 → {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)} ⊆ 𝐵)
989, 97ssfid 9299 . . . . 5 (𝜑 → {𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)} ∈ Fin)
99 hashcl 14392 . . . . 5 ({𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)} ∈ Fin → (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}) ∈ ℕ0)
10098, 99syl 17 . . . 4 (𝜑 → (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}) ∈ ℕ0)
101100nn0red 12586 . . 3 (𝜑 → (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}) ∈ ℝ)
10214nnred 12279 . . 3 (𝜑 → ((od‘𝐺)‘𝐴) ∈ ℝ)
103101, 102letri3d 11401 . 2 (𝜑 → ((♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}) = ((od‘𝐺)‘𝐴) ↔ ((♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}) ≤ ((od‘𝐺)‘𝐴) ∧ ((od‘𝐺)‘𝐴) ≤ (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}))))
10495, 103mpbird 257 1 (𝜑 → (♯‘{𝑥𝐵 ∣ (((od‘𝐺)‘𝐴) 𝑥) = (0g𝐺)}) = ((od‘𝐺)‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  ifcif 4531   class class class wbr 5148  cmpt 5231  ran crn 5690   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Fincfn 8984  0cc0 11153   · cmul 11158  cle 11294  cn 12264  0cn0 12524  cz 12611  chash 14366  Basecbs 17245  0gc0g 17486  Grpcgrp 18964  .gcmg 19098  odcod 19557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-acn 9980  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-dvds 16288  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-od 19561
This theorem is referenced by:  unitscyglem2  42178
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