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Theorem subgmulg 17829
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 2760 . . . . . 6 (0g𝐺) = (0g𝐺)
31, 2subg0 17821 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
433ad2ant1 1128 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (0g𝐺) = (0g𝐻))
54ifeq1d 4248 . . 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 2760 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
71, 6ressplusg 16215 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
873ad2ant1 1128 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (+g𝐺) = (+g𝐻))
98seqeq2d 13022 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋})))
109adantr 472 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) → seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋})))
1110fveq1d 6355 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) → (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁) = (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁))
1211ifeq1d 4248 . . . . 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 1132 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑁 ∈ ℤ)
1413zred 11694 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑁 ∈ ℝ)
15 0re 10252 . . . . . . . . . . . 12 0 ∈ ℝ
16 axlttri 10321 . . . . . . . . . . . 12 ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 < 0 ↔ ¬ (𝑁 = 0 ∨ 0 < 𝑁)))
1714, 15, 16sylancl 697 . . . . . . . . . . 11 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 < 0 ↔ ¬ (𝑁 = 0 ∨ 0 < 𝑁)))
18 ioran 512 . . . . . . . . . . 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 503 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → 𝑁 < 0)
21 simpl1 1228 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑆 ∈ (SubGrp‘𝐺))
2213adantr 472 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑁 ∈ ℤ)
2322znegcld 11696 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℤ)
2414lt0neg1d 10809 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 < 0 ↔ 0 < -𝑁))
2524biimpa 502 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 0 < -𝑁)
26 elnnz 11599 . . . . . . . . . . . . 13 (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
2723, 25, 26sylanbrc 701 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℕ)
28 eqid 2760 . . . . . . . . . . . . . . . 16 (Base‘𝐺) = (Base‘𝐺)
2928subgss 17816 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
30293ad2ant1 1128 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑆 ⊆ (Base‘𝐺))
31 simp3 1133 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑋𝑆)
3230, 31sseldd 3745 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐺))
3332adantr 472 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑋 ∈ (Base‘𝐺))
34 subgmulgcl.t . . . . . . . . . . . . 13 · = (.g𝐺)
35 eqid 2760 . . . . . . . . . . . . 13 seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐺), (ℕ × {𝑋}))
3628, 6, 34, 35mulgnn 17768 . . . . . . . . . . . 12 ((-𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐺)) → (-𝑁 · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁))
3727, 33, 36syl2anc 696 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁))
3831adantr 472 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑋𝑆)
3934subgmulgcl 17828 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ -𝑁 ∈ ℤ ∧ 𝑋𝑆) → (-𝑁 · 𝑋) ∈ 𝑆)
4021, 23, 38, 39syl3anc 1477 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) ∈ 𝑆)
4137, 40eqeltrrd 2840 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁) ∈ 𝑆)
42 eqid 2760 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
43 eqid 2760 . . . . . . . . . . 11 (invg𝐻) = (invg𝐻)
441, 42, 43subginv 17822 . . . . . . . . . 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 696 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))
4620, 45syldan 488 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))
479adantr 472 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋})))
4847fveq1d 6355 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁) = (seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁))
4948fveq2d 6357 . . . . . . . 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 2794 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))
5150anassrs 683 . . . . . 6 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))
5251ifeq2da 4261 . . . . 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 2794 . . . 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 4261 . . 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 2794 . 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 17764 . . 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 696 . 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 17819 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
59583ad2ant1 1128 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑆 = (Base‘𝐻))
6031, 59eleqtrd 2841 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐻))
61 eqid 2760 . . . 4 (Base‘𝐻) = (Base‘𝐻)
62 eqid 2760 . . . 4 (+g𝐻) = (+g𝐻)
63 eqid 2760 . . . 4 (0g𝐻) = (0g𝐻)
64 subgmulg.t . . . 4 = (.g𝐻)
65 eqid 2760 . . . 4 seq1((+g𝐻), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋}))
6661, 62, 63, 43, 64, 65mulgval 17764 . . 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 696 . 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 2804 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 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wss 3715  ifcif 4230  {csn 4321   class class class wbr 4804   × cxp 5264  cfv 6049  (class class class)co 6814  cr 10147  0cc0 10148  1c1 10149   < clt 10286  -cneg 10479  cn 11232  cz 11589  seqcseq 13015  Basecbs 16079  s cress 16080  +gcplusg 16163  0gc0g 16322  invgcminusg 17644  .gcmg 17761  SubGrpcsubg 17809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-seq 13016  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-ress 16087  df-plusg 16176  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-minusg 17647  df-mulg 17762  df-subg 17812
This theorem is referenced by:  cycsubgcyg  18522  ablfac2  18708  zringmulg  20048  zringcyg  20061  remulg  20175  rezh  30345
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