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Theorem subgmulg 19171
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 2735 . . . . . 6 (0g𝐺) = (0g𝐺)
31, 2subg0 19163 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
433ad2ant1 1132 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (0g𝐺) = (0g𝐻))
54ifeq1d 4550 . . 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 2735 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
71, 6ressplusg 17336 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
873ad2ant1 1132 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (+g𝐺) = (+g𝐻))
98seqeq2d 14046 . . . . . . . 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 6909 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) → (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁) = (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁))
1211ifeq1d 4550 . . . . 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 1136 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑁 ∈ ℤ)
1413zred 12720 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑁 ∈ ℝ)
15 0re 11261 . . . . . . . . . . . 12 0 ∈ ℝ
16 axlttri 11330 . . . . . . . . . . . 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 985 . . . . . . . . . . 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 1190 . . . . . . . . . 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 12722 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℤ)
2414lt0neg1d 11830 . . . . . . . . . . . . . 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 12621 . . . . . . . . . . . . 13 (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
2723, 25, 26sylanbrc 583 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℕ)
28 eqid 2735 . . . . . . . . . . . . . . . 16 (Base‘𝐺) = (Base‘𝐺)
2928subgss 19158 . . . . . . . . . . . . . . 15 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
30293ad2ant1 1132 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑆 ⊆ (Base‘𝐺))
31 simp3 1137 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑋𝑆)
3230, 31sseldd 3996 . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . 13 seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐺), (ℕ × {𝑋}))
3628, 6, 34, 35mulgnn 19106 . . . . . . . . . . . 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 19170 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ -𝑁 ∈ ℤ ∧ 𝑋𝑆) → (-𝑁 · 𝑋) ∈ 𝑆)
4021, 23, 38, 39syl3anc 1370 . . . . . . . . . . 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 2735 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
43 eqid 2735 . . . . . . . . . . 11 (invg𝐻) = (invg𝐻)
441, 42, 43subginv 19164 . . . . . . . . . 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 6909 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁) = (seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁))
4948fveq2d 6911 . . . . . . . 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 2775 . . . . . . 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 4563 . . . . 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 2775 . . . 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 4563 . . 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 2775 . 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 19102 . . 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 19161 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
59583ad2ant1 1132 . . . 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 2735 . . . 4 (Base‘𝐻) = (Base‘𝐻)
62 eqid 2735 . . . 4 (+g𝐻) = (+g𝐻)
63 eqid 2735 . . . 4 (0g𝐻) = (0g𝐻)
64 subgmulg.t . . . 4 = (.g𝐻)
65 eqid 2735 . . . 4 seq1((+g𝐻), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋}))
6661, 62, 63, 43, 64, 65mulgval 19102 . . 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 2785 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 847  w3a 1086   = wceq 1537  wcel 2106  wss 3963  ifcif 4531  {csn 4631   class class class wbr 5148   × cxp 5687  cfv 6563  (class class class)co 7431  cr 11152  0cc0 11153  1c1 11154   < clt 11293  -cneg 11491  cn 12264  cz 12611  seqcseq 14039  Basecbs 17245  s cress 17274  +gcplusg 17298  0gc0g 17486  invgcminusg 18965  .gcmg 19098  SubGrpcsubg 19151
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-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  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
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-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-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-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-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-seq 14040  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-mulg 19099  df-subg 19154
This theorem is referenced by:  cycsubgcyg  19934  ablfac2  20124  zringmulg  21485  zringcyg  21498  remulg  21643  subgmulgcld  33031  rezh  33932
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