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Theorem frgpcyg 20720
Description: A free group is cyclic iff it has zero or one generator. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 18-Apr-2021.)
Hypothesis
Ref Expression
frgpcyg.g 𝐺 = (freeGrp‘𝐼)
Assertion
Ref Expression
frgpcyg (𝐼 ≼ 1o𝐺 ∈ CycGrp)

Proof of Theorem frgpcyg
Dummy variables 𝑓 𝑔 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brdom2 8539 . . 3 (𝐼 ≼ 1o ↔ (𝐼 ≺ 1o𝐼 ≈ 1o))
2 sdom1 8718 . . . . 5 (𝐼 ≺ 1o𝐼 = ∅)
3 frgpcyg.g . . . . . . 7 𝐺 = (freeGrp‘𝐼)
4 fveq2 6670 . . . . . . 7 (𝐼 = ∅ → (freeGrp‘𝐼) = (freeGrp‘∅))
53, 4syl5eq 2868 . . . . . 6 (𝐼 = ∅ → 𝐺 = (freeGrp‘∅))
6 0ex 5211 . . . . . . . 8 ∅ ∈ V
7 eqid 2821 . . . . . . . . 9 (freeGrp‘∅) = (freeGrp‘∅)
87frgpgrp 18888 . . . . . . . 8 (∅ ∈ V → (freeGrp‘∅) ∈ Grp)
96, 8ax-mp 5 . . . . . . 7 (freeGrp‘∅) ∈ Grp
10 eqid 2821 . . . . . . . 8 (Base‘(freeGrp‘∅)) = (Base‘(freeGrp‘∅))
117, 100frgp 18905 . . . . . . 7 (Base‘(freeGrp‘∅)) ≈ 1o
12100cyg 19013 . . . . . . 7 (((freeGrp‘∅) ∈ Grp ∧ (Base‘(freeGrp‘∅)) ≈ 1o) → (freeGrp‘∅) ∈ CycGrp)
139, 11, 12mp2an 690 . . . . . 6 (freeGrp‘∅) ∈ CycGrp
145, 13eqeltrdi 2921 . . . . 5 (𝐼 = ∅ → 𝐺 ∈ CycGrp)
152, 14sylbi 219 . . . 4 (𝐼 ≺ 1o𝐺 ∈ CycGrp)
16 eqid 2821 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
17 eqid 2821 . . . . 5 (.g𝐺) = (.g𝐺)
18 relen 8514 . . . . . . 7 Rel ≈
1918brrelex1i 5608 . . . . . 6 (𝐼 ≈ 1o𝐼 ∈ V)
203frgpgrp 18888 . . . . . 6 (𝐼 ∈ V → 𝐺 ∈ Grp)
2119, 20syl 17 . . . . 5 (𝐼 ≈ 1o𝐺 ∈ Grp)
22 eqid 2821 . . . . . . . 8 ( ~FG𝐼) = ( ~FG𝐼)
23 eqid 2821 . . . . . . . 8 (varFGrp𝐼) = (varFGrp𝐼)
2422, 23, 3, 16vrgpf 18894 . . . . . . 7 (𝐼 ∈ V → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
2519, 24syl 17 . . . . . 6 (𝐼 ≈ 1o → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
26 en1uniel 8581 . . . . . 6 (𝐼 ≈ 1o 𝐼𝐼)
2725, 26ffvelrnd 6852 . . . . 5 (𝐼 ≈ 1o → ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺))
28 zringgrp 20622 . . . . . . . . 9 ring ∈ Grp
2919uniexd 7468 . . . . . . . . . . 11 (𝐼 ≈ 1o 𝐼 ∈ V)
30 1zzd 12014 . . . . . . . . . . 11 (𝐼 ≈ 1o → 1 ∈ ℤ)
3129, 30fsnd 6657 . . . . . . . . . 10 (𝐼 ≈ 1o → {⟨ 𝐼, 1⟩}:{ 𝐼}⟶ℤ)
32 en1b 8577 . . . . . . . . . . . 12 (𝐼 ≈ 1o𝐼 = { 𝐼})
3332biimpi 218 . . . . . . . . . . 11 (𝐼 ≈ 1o𝐼 = { 𝐼})
3433feq2d 6500 . . . . . . . . . 10 (𝐼 ≈ 1o → ({⟨ 𝐼, 1⟩}:𝐼⟶ℤ ↔ {⟨ 𝐼, 1⟩}:{ 𝐼}⟶ℤ))
3531, 34mpbird 259 . . . . . . . . 9 (𝐼 ≈ 1o → {⟨ 𝐼, 1⟩}:𝐼⟶ℤ)
36 zringbas 20623 . . . . . . . . . 10 ℤ = (Base‘ℤring)
373, 36, 23frgpup3 18904 . . . . . . . . 9 ((ℤring ∈ Grp ∧ 𝐼 ∈ V ∧ {⟨ 𝐼, 1⟩}:𝐼⟶ℤ) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
3828, 19, 35, 37mp3an2i 1462 . . . . . . . 8 (𝐼 ≈ 1o → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
3938adantr 483 . . . . . . 7 ((𝐼 ≈ 1o𝑥 ∈ (Base‘𝐺)) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
40 reurex 3431 . . . . . . 7 (∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
4139, 40syl 17 . . . . . 6 ((𝐼 ≈ 1o𝑥 ∈ (Base‘𝐺)) → ∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
42 fveq1 6669 . . . . . . . . . 10 ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = ({⟨ 𝐼, 1⟩}‘ 𝐼))
4325, 26fvco3d 6761 . . . . . . . . . . 11 (𝐼 ≈ 1o → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
44 1z 12013 . . . . . . . . . . . 12 1 ∈ ℤ
45 fvsng 6942 . . . . . . . . . . . 12 (( 𝐼 ∈ V ∧ 1 ∈ ℤ) → ({⟨ 𝐼, 1⟩}‘ 𝐼) = 1)
4629, 44, 45sylancl 588 . . . . . . . . . . 11 (𝐼 ≈ 1o → ({⟨ 𝐼, 1⟩}‘ 𝐼) = 1)
4743, 46eqeq12d 2837 . . . . . . . . . 10 (𝐼 ≈ 1o → (((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = ({⟨ 𝐼, 1⟩}‘ 𝐼) ↔ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
4842, 47syl5ib 246 . . . . . . . . 9 (𝐼 ≈ 1o → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
4948ad2antrr 724 . . . . . . . 8 (((𝐼 ≈ 1o𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
5016, 36ghmf 18362 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐺 GrpHom ℤring) → 𝑓:(Base‘𝐺)⟶ℤ)
5150ad2antrl 726 . . . . . . . . . . . 12 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓:(Base‘𝐺)⟶ℤ)
5251ffvelrnda 6851 . . . . . . . . . . 11 (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑥) ∈ ℤ)
5352an32s 650 . . . . . . . . . 10 (((𝐼 ≈ 1o𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑓𝑥) ∈ ℤ)
54 mptresid 5918 . . . . . . . . . . . . . 14 ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ 𝑥)
553, 16, 23frgpup3 18904 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝐼 ∈ V ∧ (varFGrp𝐼):𝐼⟶(Base‘𝐺)) → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
5621, 19, 25, 55syl3anc 1367 . . . . . . . . . . . . . . . . 17 (𝐼 ≈ 1o → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
57 reurmo 3433 . . . . . . . . . . . . . . . . 17 (∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
5856, 57syl 17 . . . . . . . . . . . . . . . 16 (𝐼 ≈ 1o → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
5958adantr 483 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
6021adantr 483 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝐺 ∈ Grp)
6116idghm 18373 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → ( I ↾ (Base‘𝐺)) ∈ (𝐺 GrpHom 𝐺))
6260, 61syl 17 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ( I ↾ (Base‘𝐺)) ∈ (𝐺 GrpHom 𝐺))
6325adantr 483 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
64 fcoi2 6553 . . . . . . . . . . . . . . . 16 ((varFGrp𝐼):𝐼⟶(Base‘𝐺) → (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)) = (varFGrp𝐼))
6563, 64syl 17 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)) = (varFGrp𝐼))
6651feqmptd 6733 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑓𝑥)))
67 eqidd 2822 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
68 oveq1 7163 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑓𝑥) → (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
6952, 66, 67, 68fmptco 6891 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ 𝑓) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
7027adantr 483 . . . . . . . . . . . . . . . . . 18 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺))
71 eqid 2821 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
7217, 71, 16mulgghm2 20644 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺)) → (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (ℤring GrpHom 𝐺))
7360, 70, 72syl2anc 586 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (ℤring GrpHom 𝐺))
74 simprl 769 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓 ∈ (𝐺 GrpHom ℤring))
75 ghmco 18378 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (ℤring GrpHom 𝐺) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ 𝑓) ∈ (𝐺 GrpHom 𝐺))
7673, 74, 75syl2anc 586 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ 𝑓) ∈ (𝐺 GrpHom 𝐺))
7769, 76eqeltrrd 2914 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (𝐺 GrpHom 𝐺))
7833adantr 483 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝐼 = { 𝐼})
7978eleq2d 2898 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼𝑦 ∈ { 𝐼}))
80 simprr 771 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)
8180oveq1d 7171 . . . . . . . . . . . . . . . . . . . . 21 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = (1(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
8216, 17mulg1 18235 . . . . . . . . . . . . . . . . . . . . . 22 (((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺) → (1(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼))
8370, 82syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (1(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼))
8481, 83eqtrd 2856 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼))
85 elsni 4584 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ { 𝐼} → 𝑦 = 𝐼)
8685fveq2d 6674 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ { 𝐼} → ((varFGrp𝐼)‘𝑦) = ((varFGrp𝐼)‘ 𝐼))
8786fveq2d 6674 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ { 𝐼} → (𝑓‘((varFGrp𝐼)‘𝑦)) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
8887oveq1d 7171 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ { 𝐼} → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
8988, 86eqeq12d 2837 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ { 𝐼} → (((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦) ↔ ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼)))
9084, 89syl5ibrcom 249 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦 ∈ { 𝐼} → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦)))
9179, 90sylbid 242 . . . . . . . . . . . . . . . . . 18 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼 → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦)))
9291imp 409 . . . . . . . . . . . . . . . . 17 (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑦𝐼) → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦))
9392mpteq2dva 5161 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼 ↦ ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑦𝐼 ↦ ((varFGrp𝐼)‘𝑦)))
9463ffvelrnda 6851 . . . . . . . . . . . . . . . . 17 (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑦𝐼) → ((varFGrp𝐼)‘𝑦) ∈ (Base‘𝐺))
9563feqmptd 6733 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (varFGrp𝐼) = (𝑦𝐼 ↦ ((varFGrp𝐼)‘𝑦)))
96 eqidd 2822 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
97 fveq2 6670 . . . . . . . . . . . . . . . . . 18 (𝑥 = ((varFGrp𝐼)‘𝑦) → (𝑓𝑥) = (𝑓‘((varFGrp𝐼)‘𝑦)))
9897oveq1d 7171 . . . . . . . . . . . . . . . . 17 (𝑥 = ((varFGrp𝐼)‘𝑦) → ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
9994, 95, 96, 98fmptco 6891 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)) = (𝑦𝐼 ↦ ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
10093, 99, 953eqtr4d 2866 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)) = (varFGrp𝐼))
101 coeq1 5728 . . . . . . . . . . . . . . . . 17 (𝑔 = ( I ↾ (Base‘𝐺)) → (𝑔 ∘ (varFGrp𝐼)) = (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)))
102101eqeq1d 2823 . . . . . . . . . . . . . . . 16 (𝑔 = ( I ↾ (Base‘𝐺)) → ((𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) ↔ (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)) = (varFGrp𝐼)))
103 coeq1 5728 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → (𝑔 ∘ (varFGrp𝐼)) = ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)))
104103eqeq1d 2823 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → ((𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)) = (varFGrp𝐼)))
105102, 104rmoi 3875 . . . . . . . . . . . . . . 15 ((∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) ∧ (( I ↾ (Base‘𝐺)) ∈ (𝐺 GrpHom 𝐺) ∧ (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)) = (varFGrp𝐼)) ∧ ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (𝐺 GrpHom 𝐺) ∧ ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)) = (varFGrp𝐼))) → ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
10659, 62, 65, 77, 100, 105syl122anc 1375 . . . . . . . . . . . . . 14 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
10754, 106syl5eqr 2870 . . . . . . . . . . . . 13 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
108 mpteqb 6787 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (Base‘𝐺)𝑥 ∈ (Base‘𝐺) → ((𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ↔ ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
109 id 22 . . . . . . . . . . . . . 14 (𝑥 ∈ (Base‘𝐺) → 𝑥 ∈ (Base‘𝐺))
110108, 109mprg 3152 . . . . . . . . . . . . 13 ((𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ↔ ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
111107, 110sylib 220 . . . . . . . . . . . 12 ((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
112111r19.21bi 3208 . . . . . . . . . . 11 (((𝐼 ≈ 1o ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
113112an32s 650 . . . . . . . . . 10 (((𝐼 ≈ 1o𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
11468rspceeqv 3638 . . . . . . . . . 10 (((𝑓𝑥) ∈ ℤ ∧ 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
11553, 113, 114syl2anc 586 . . . . . . . . 9 (((𝐼 ≈ 1o𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
116115expr 459 . . . . . . . 8 (((𝐼 ≈ 1o𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1 → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
11749, 116syld 47 . . . . . . 7 (((𝐼 ≈ 1o𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
118117rexlimdva 3284 . . . . . 6 ((𝐼 ≈ 1o𝑥 ∈ (Base‘𝐺)) → (∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
11941, 118mpd 15 . . . . 5 ((𝐼 ≈ 1o𝑥 ∈ (Base‘𝐺)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
12016, 17, 21, 27, 119iscygd 19006 . . . 4 (𝐼 ≈ 1o𝐺 ∈ CycGrp)
12115, 120jaoi 853 . . 3 ((𝐼 ≺ 1o𝐼 ≈ 1o) → 𝐺 ∈ CycGrp)
1221, 121sylbi 219 . 2 (𝐼 ≼ 1o𝐺 ∈ CycGrp)
123 cygabl 19010 . . 3 (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
1243frgpnabl 18995 . . . . 5 (1o𝐼 → ¬ 𝐺 ∈ Abel)
125124con2i 141 . . . 4 (𝐺 ∈ Abel → ¬ 1o𝐼)
126 ablgrp 18911 . . . . . 6 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
127 eqid 2821 . . . . . . 7 (0g𝐺) = (0g𝐺)
12816, 127grpidcl 18131 . . . . . 6 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
1293, 16elbasfv 16544 . . . . . 6 ((0g𝐺) ∈ (Base‘𝐺) → 𝐼 ∈ V)
130126, 128, 1293syl 18 . . . . 5 (𝐺 ∈ Abel → 𝐼 ∈ V)
131 1onn 8265 . . . . . 6 1o ∈ ω
132 nnfi 8711 . . . . . 6 (1o ∈ ω → 1o ∈ Fin)
133131, 132ax-mp 5 . . . . 5 1o ∈ Fin
134 fidomtri2 9423 . . . . 5 ((𝐼 ∈ V ∧ 1o ∈ Fin) → (𝐼 ≼ 1o ↔ ¬ 1o𝐼))
135130, 133, 134sylancl 588 . . . 4 (𝐺 ∈ Abel → (𝐼 ≼ 1o ↔ ¬ 1o𝐼))
136125, 135mpbird 259 . . 3 (𝐺 ∈ Abel → 𝐼 ≼ 1o)
137123, 136syl 17 . 2 (𝐺 ∈ CycGrp → 𝐼 ≼ 1o)
138122, 137impbii 211 1 (𝐼 ≼ 1o𝐺 ∈ CycGrp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843   = wceq 1537  wcel 2114  wral 3138  wrex 3139  ∃!wreu 3140  ∃*wrmo 3141  Vcvv 3494  c0 4291  {csn 4567  cop 4573   cuni 4838   class class class wbr 5066  cmpt 5146   I cid 5459  cres 5557  ccom 5559  wf 6351  cfv 6355  (class class class)co 7156  ωcom 7580  1oc1o 8095  cen 8506  cdom 8507  csdm 8508  Fincfn 8509  1c1 10538  cz 11982  Basecbs 16483  0gc0g 16713  Grpcgrp 18103  .gcmg 18224   GrpHom cghm 18355   ~FG cefg 18832  freeGrpcfrgp 18833  varFGrpcvrgp 18834  Abelcabl 18907  CycGrpccyg 18996  ringzring 20617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-addf 10616  ax-mulf 10617
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-ec 8291  df-qs 8295  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-inf 8907  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-xnn0 11969  df-z 11983  df-dec 12100  df-uz 12245  df-rp 12391  df-fz 12894  df-fzo 13035  df-seq 13371  df-hash 13692  df-word 13863  df-lsw 13915  df-concat 13923  df-s1 13950  df-substr 14003  df-pfx 14033  df-splice 14112  df-reverse 14121  df-s2 14210  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-starv 16580  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-unif 16588  df-0g 16715  df-gsum 16716  df-imas 16781  df-qus 16782  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-frmd 18014  df-vrmd 18015  df-grp 18106  df-minusg 18107  df-mulg 18225  df-subg 18276  df-ghm 18356  df-efg 18835  df-frgp 18836  df-vrgp 18837  df-cmn 18908  df-abl 18909  df-cyg 18997  df-mgp 19240  df-ur 19252  df-ring 19299  df-cring 19300  df-subrg 19533  df-cnfld 20546  df-zring 20618
This theorem is referenced by: (None)
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