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Theorem subgmulg 17524
Description: A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
Hypotheses
Ref Expression
subgmulgcl.t · = (.g𝐺)
subgmulg.h 𝐻 = (𝐺s 𝑆)
subgmulg.t = (.g𝐻)
Assertion
Ref Expression
subgmulg ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 · 𝑋) = (𝑁 𝑋))

Proof of Theorem subgmulg
StepHypRef Expression
1 subgmulg.h . . . . . 6 𝐻 = (𝐺s 𝑆)
2 eqid 2626 . . . . . 6 (0g𝐺) = (0g𝐺)
31, 2subg0 17516 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
433ad2ant1 1080 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (0g𝐺) = (0g𝐻))
54ifeq1d 4081 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → if(𝑁 = 0, (0g𝐺), if(0 < 𝑁, (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))) = if(𝑁 = 0, (0g𝐻), if(0 < 𝑁, (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))))
6 eqid 2626 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
71, 6ressplusg 15909 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
873ad2ant1 1080 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (+g𝐺) = (+g𝐻))
98seqeq2d 12745 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋})))
109adantr 481 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) → seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋})))
1110fveq1d 6152 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) → (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁) = (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁))
1211ifeq1d 4081 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁))) = if(0 < 𝑁, (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁))))
13 simp2 1060 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑁 ∈ ℤ)
1413zred 11426 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑁 ∈ ℝ)
15 0re 9985 . . . . . . . . . . . 12 0 ∈ ℝ
16 axlttri 10054 . . . . . . . . . . . 12 ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 < 0 ↔ ¬ (𝑁 = 0 ∨ 0 < 𝑁)))
1714, 15, 16sylancl 693 . . . . . . . . . . 11 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 < 0 ↔ ¬ (𝑁 = 0 ∨ 0 < 𝑁)))
18 ioran 511 . . . . . . . . . . 11 (¬ (𝑁 = 0 ∨ 0 < 𝑁) ↔ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁))
1917, 18syl6bb 276 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 < 0 ↔ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)))
2019biimpar 502 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → 𝑁 < 0)
21 simpl1 1062 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑆 ∈ (SubGrp‘𝐺))
2213adantr 481 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑁 ∈ ℤ)
2322znegcld 11428 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℤ)
2414lt0neg1d 10542 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 < 0 ↔ 0 < -𝑁))
2524biimpa 501 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 0 < -𝑁)
26 elnnz 11332 . . . . . . . . . . . . 13 (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
2723, 25, 26sylanbrc 697 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℕ)
28 eqid 2626 . . . . . . . . . . . . . . . 16 (Base‘𝐺) = (Base‘𝐺)
2928subgss 17511 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
30293ad2ant1 1080 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑆 ⊆ (Base‘𝐺))
31 simp3 1061 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑋𝑆)
3230, 31sseldd 3589 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐺))
3332adantr 481 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑋 ∈ (Base‘𝐺))
34 subgmulgcl.t . . . . . . . . . . . . 13 · = (.g𝐺)
35 eqid 2626 . . . . . . . . . . . . 13 seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐺), (ℕ × {𝑋}))
3628, 6, 34, 35mulgnn 17463 . . . . . . . . . . . 12 ((-𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐺)) → (-𝑁 · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁))
3727, 33, 36syl2anc 692 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁))
3831adantr 481 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑋𝑆)
3934subgmulgcl 17523 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ -𝑁 ∈ ℤ ∧ 𝑋𝑆) → (-𝑁 · 𝑋) ∈ 𝑆)
4021, 23, 38, 39syl3anc 1323 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) ∈ 𝑆)
4137, 40eqeltrrd 2705 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁) ∈ 𝑆)
42 eqid 2626 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
43 eqid 2626 . . . . . . . . . . 11 (invg𝐻) = (invg𝐻)
441, 42, 43subginv 17517 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁) ∈ 𝑆) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))
4521, 41, 44syl2anc 692 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))
4620, 45syldan 487 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))
479adantr 481 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋})))
4847fveq1d 6152 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁) = (seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁))
4948fveq2d 6154 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ((invg𝐻)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))
5046, 49eqtrd 2660 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))
5150anassrs 679 . . . . . 6 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))
5251ifeq2da 4094 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁))) = if(0 < 𝑁, (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁))))
5312, 52eqtrd 2660 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁))) = if(0 < 𝑁, (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁))))
5453ifeq2da 4094 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → if(𝑁 = 0, (0g𝐻), if(0 < 𝑁, (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))) = if(𝑁 = 0, (0g𝐻), if(0 < 𝑁, (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))))
555, 54eqtrd 2660 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → if(𝑁 = 0, (0g𝐺), if(0 < 𝑁, (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))) = if(𝑁 = 0, (0g𝐻), if(0 < 𝑁, (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))))
5628, 6, 2, 42, 34, 35mulgval 17459 . . 3 ((𝑁 ∈ ℤ ∧ 𝑋 ∈ (Base‘𝐺)) → (𝑁 · 𝑋) = if(𝑁 = 0, (0g𝐺), if(0 < 𝑁, (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))))
5713, 32, 56syl2anc 692 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 · 𝑋) = if(𝑁 = 0, (0g𝐺), if(0 < 𝑁, (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))))
581subgbas 17514 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
59583ad2ant1 1080 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑆 = (Base‘𝐻))
6031, 59eleqtrd 2706 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐻))
61 eqid 2626 . . . 4 (Base‘𝐻) = (Base‘𝐻)
62 eqid 2626 . . . 4 (+g𝐻) = (+g𝐻)
63 eqid 2626 . . . 4 (0g𝐻) = (0g𝐻)
64 subgmulg.t . . . 4 = (.g𝐻)
65 eqid 2626 . . . 4 seq1((+g𝐻), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋}))
6661, 62, 63, 43, 64, 65mulgval 17459 . . 3 ((𝑁 ∈ ℤ ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑁 𝑋) = if(𝑁 = 0, (0g𝐻), if(0 < 𝑁, (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))))
6713, 60, 66syl2anc 692 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 𝑋) = if(𝑁 = 0, (0g𝐻), if(0 < 𝑁, (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))))
6855, 57, 673eqtr4d 2670 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 · 𝑋) = (𝑁 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1992  wss 3560  ifcif 4063  {csn 4153   class class class wbr 4618   × cxp 5077  cfv 5850  (class class class)co 6605  cr 9880  0cc0 9881  1c1 9882   < clt 10019  -cneg 10212  cn 10965  cz 11322  seqcseq 12738  Basecbs 15776  s cress 15777  +gcplusg 15857  0gc0g 16016  invgcminusg 17339  .gcmg 17456  SubGrpcsubg 17504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-seq 12739  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-minusg 17342  df-mulg 17457  df-subg 17507
This theorem is referenced by:  cycsubgcyg  18218  ablfac2  18404  zringmulg  19740  zringcyg  19753  remulg  19867  rezh  29789
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