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Theorem subgmulg 19005
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 2733 . . . . . 6 (0g𝐺) = (0g𝐺)
31, 2subg0 18997 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → (0g𝐺) = (0g𝐻))
433ad2ant1 1134 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (0g𝐺) = (0g𝐻))
54ifeq1d 4543 . . 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 2733 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
71, 6ressplusg 17222 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
873ad2ant1 1134 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (+g𝐺) = (+g𝐻))
98seqeq2d 13960 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋})))
109adantr 482 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) → seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋})))
1110fveq1d 6883 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) → (seq1((+g𝐺), (ℕ × {𝑋}))‘𝑁) = (seq1((+g𝐻), (ℕ × {𝑋}))‘𝑁))
1211ifeq1d 4543 . . . . 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 12653 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑁 ∈ ℝ)
15 0re 11203 . . . . . . . . . . . 12 0 ∈ ℝ
16 axlttri 11272 . . . . . . . . . . . 12 ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 < 0 ↔ ¬ (𝑁 = 0 ∨ 0 < 𝑁)))
1714, 15, 16sylancl 587 . . . . . . . . . . 11 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 < 0 ↔ ¬ (𝑁 = 0 ∨ 0 < 𝑁)))
18 ioran 983 . . . . . . . . . . 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 479 . . . . . . . . 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 482 . . . . . . . . . . . . . 14 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑁 ∈ ℤ)
2322znegcld 12655 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℤ)
2414lt0neg1d 11770 . . . . . . . . . . . . . 14 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 < 0 ↔ 0 < -𝑁))
2524biimpa 478 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 0 < -𝑁)
26 elnnz 12555 . . . . . . . . . . . . 13 (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
2723, 25, 26sylanbrc 584 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → -𝑁 ∈ ℕ)
28 eqid 2733 . . . . . . . . . . . . . . . 16 (Base‘𝐺) = (Base‘𝐺)
2928subgss 18992 . . . . . . . . . . . . . . 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 3981 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐺))
3332adantr 482 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑋 ∈ (Base‘𝐺))
34 subgmulgcl.t . . . . . . . . . . . . 13 · = (.g𝐺)
35 eqid 2733 . . . . . . . . . . . . 13 seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐺), (ℕ × {𝑋}))
3628, 6, 34, 35mulgnn 18943 . . . . . . . . . . . 12 ((-𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐺)) → (-𝑁 · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁))
3727, 33, 36syl2anc 585 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) = (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁))
3831adantr 482 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → 𝑋𝑆)
3934subgmulgcl 19004 . . . . . . . . . . . 12 ((𝑆 ∈ (SubGrp‘𝐺) ∧ -𝑁 ∈ ℤ ∧ 𝑋𝑆) → (-𝑁 · 𝑋) ∈ 𝑆)
4021, 23, 38, 39syl3anc 1372 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → (-𝑁 · 𝑋) ∈ 𝑆)
4137, 40eqeltrrd 2835 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁) ∈ 𝑆)
42 eqid 2733 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
43 eqid 2733 . . . . . . . . . . 11 (invg𝐻) = (invg𝐻)
441, 42, 43subginv 18998 . . . . . . . . . 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 585 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ 𝑁 < 0) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))
4620, 45syldan 592 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)))
479adantr 482 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → seq1((+g𝐺), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋})))
4847fveq1d 6883 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → (seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁) = (seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁))
4948fveq2d 6885 . . . . . . . 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 2773 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ (¬ 𝑁 = 0 ∧ ¬ 0 < 𝑁)) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))
5150anassrs 469 . . . . . 6 ((((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑋}))‘-𝑁)) = ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑋}))‘-𝑁)))
5251ifeq2da 4556 . . . . 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 2773 . . . 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 4556 . . 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 2773 . 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 18939 . . 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 585 . 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 18995 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
59583ad2ant1 1134 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑆 = (Base‘𝐻))
6031, 59eleqtrd 2836 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → 𝑋 ∈ (Base‘𝐻))
61 eqid 2733 . . . 4 (Base‘𝐻) = (Base‘𝐻)
62 eqid 2733 . . . 4 (+g𝐻) = (+g𝐻)
63 eqid 2733 . . . 4 (0g𝐻) = (0g𝐻)
64 subgmulg.t . . . 4 = (.g𝐻)
65 eqid 2733 . . . 4 seq1((+g𝐻), (ℕ × {𝑋})) = seq1((+g𝐻), (ℕ × {𝑋}))
6661, 62, 63, 43, 64, 65mulgval 18939 . . 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 585 . 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 2783 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋𝑆) → (𝑁 · 𝑋) = (𝑁 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wss 3946  ifcif 4524  {csn 4624   class class class wbr 5144   × cxp 5670  cfv 6535  (class class class)co 7396  cr 11096  0cc0 11097  1c1 11098   < clt 11235  -cneg 11432  cn 12199  cz 12545  seqcseq 13953  Basecbs 17131  s cress 17160  +gcplusg 17184  0gc0g 17372  invgcminusg 18807  .gcmg 18935  SubGrpcsubg 18985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-mulcom 11161  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173  ax-pre-mulgt0 11174
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7843  df-1st 7962  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-er 8691  df-en 8928  df-dom 8929  df-sdom 8930  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-sub 11433  df-neg 11434  df-nn 12200  df-2 12262  df-n0 12460  df-z 12546  df-uz 12810  df-fz 13472  df-seq 13954  df-sets 17084  df-slot 17102  df-ndx 17114  df-base 17132  df-ress 17161  df-plusg 17197  df-0g 17374  df-mgm 18548  df-sgrp 18597  df-mnd 18613  df-grp 18809  df-minusg 18810  df-mulg 18936  df-subg 18988
This theorem is referenced by:  cycsubgcyg  19752  ablfac2  19942  zringmulg  20999  zringcyg  21012  remulg  21133  rezh  32882
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