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Theorem subgmulg 19158
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 2737 . . . . . 6 (0g𝐺) = (0g𝐺)
31, 2subg0 19150 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
433ad2ant1 1134 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (0g𝐺) = (0g𝐻))
54ifeq1d 4545 . . 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 2737 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
71, 6ressplusg 17334 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
873ad2ant1 1134 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (+g𝐺) = (+g𝐻))
98seqeq2d 14049 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋})))
109adantr 480 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) → seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋})))
1110fveq1d 6908 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) → (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁) = (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁))
1211ifeq1d 4545 . . . . 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 1138 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑁 ∈ ℤ)
1413zred 12722 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑁 ∈ ℝ)
15 0re 11263 . . . . . . . . . . . 12 0 ∈ ℝ
16 axlttri 11332 . . . . . . . . . . . 12 ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 < 0 ↔ ¬ (𝑁 = 0 ∨ 0 < 𝑁)))
1714, 15, 16sylancl 586 . . . . . . . . . . 11 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 < 0 ↔ ¬ (𝑁 = 0 ∨ 0 < 𝑁)))
18 ioran 986 . . . . . . . . . . 11 (¬ (𝑁 = 0 ∨ 0 < 𝑁) ↔ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁))
1917, 18bitrdi 287 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 < 0 ↔ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)))
2019biimpar 477 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → 𝑁 < 0)
21 simpl1 1192 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑆 ∈ (SubGrp‘𝐺))
2213adantr 480 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑁 ∈ ℤ)
2322znegcld 12724 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℤ)
2414lt0neg1d 11832 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 < 0 ↔ 0 < -𝑁))
2524biimpa 476 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 0 < -𝑁)
26 elnnz 12623 . . . . . . . . . . . . 13 (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
2723, 25, 26sylanbrc 583 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℕ)
28 eqid 2737 . . . . . . . . . . . . . . . 16 (Base‘𝐺) = (Base‘𝐺)
2928subgss 19145 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
30293ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑆 ⊆ (Base‘𝐺))
31 simp3 1139 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑋𝑆)
3230, 31sseldd 3984 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐺))
3332adantr 480 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑋 ∈ (Base‘𝐺))
34 subgmulgcl.t . . . . . . . . . . . . 13 · = (.g𝐺)
35 eqid 2737 . . . . . . . . . . . . 13 seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐺), (ℕ × {𝑋}))
3628, 6, 34, 35mulgnn 19093 . . . . . . . . . . . 12 ((-𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐺)) → (-𝑁 · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁))
3727, 33, 36syl2anc 584 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁))
3831adantr 480 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑋𝑆)
3934subgmulgcl 19157 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ -𝑁 ∈ ℤ ∧ 𝑋𝑆) → (-𝑁 · 𝑋) ∈ 𝑆)
4021, 23, 38, 39syl3anc 1373 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) ∈ 𝑆)
4137, 40eqeltrrd 2842 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁) ∈ 𝑆)
42 eqid 2737 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
43 eqid 2737 . . . . . . . . . . 11 (invg𝐻) = (invg𝐻)
441, 42, 43subginv 19151 . . . . . . . . . 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 584 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))
4620, 45syldan 591 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))
479adantr 480 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋})))
4847fveq1d 6908 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁) = (seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁))
4948fveq2d 6910 . . . . . . . 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 2777 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))
5150anassrs 467 . . . . . 6 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))
5251ifeq2da 4558 . . . . 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 2777 . . . 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 4558 . . 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 2777 . 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 19089 . . 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 584 . 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 19148 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
59583ad2ant1 1134 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑆 = (Base‘𝐻))
6031, 59eleqtrd 2843 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐻))
61 eqid 2737 . . . 4 (Base‘𝐻) = (Base‘𝐻)
62 eqid 2737 . . . 4 (+g𝐻) = (+g𝐻)
63 eqid 2737 . . . 4 (0g𝐻) = (0g𝐻)
64 subgmulg.t . . . 4 = (.g𝐻)
65 eqid 2737 . . . 4 seq1((+g𝐻), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋}))
6661, 62, 63, 43, 64, 65mulgval 19089 . . 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 584 . 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 2787 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 · 𝑋) = (𝑁 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wss 3951  ifcif 4525  {csn 4626   class class class wbr 5143   × cxp 5683  cfv 6561  (class class class)co 7431  cr 11154  0cc0 11155  1c1 11156   < clt 11295  -cneg 11493  cn 12266  cz 12613  seqcseq 14042  Basecbs 17247  s cress 17274  +gcplusg 17297  0gc0g 17484  invgcminusg 18952  .gcmg 19085  SubGrpcsubg 19138
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-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-rmo 3380  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-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-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  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-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-seq 14043  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-mulg 19086  df-subg 19141
This theorem is referenced by:  cycsubgcyg  19919  ablfac2  20109  zringmulg  21467  zringcyg  21480  remulg  21625  subgmulgcld  33048  rezh  33970
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