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Theorem frgpcyg 20144
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 8153 . . 3 (𝐼 ≼ 1𝑜 ↔ (𝐼 ≺ 1𝑜𝐼 ≈ 1𝑜))
2 sdom1 8327 . . . . 5 (𝐼 ≺ 1𝑜𝐼 = ∅)
3 frgpcyg.g . . . . . . 7 𝐺 = (freeGrp‘𝐼)
4 fveq2 6353 . . . . . . 7 (𝐼 = ∅ → (freeGrp‘𝐼) = (freeGrp‘∅))
53, 4syl5eq 2806 . . . . . 6 (𝐼 = ∅ → 𝐺 = (freeGrp‘∅))
6 0ex 4942 . . . . . . . 8 ∅ ∈ V
7 eqid 2760 . . . . . . . . 9 (freeGrp‘∅) = (freeGrp‘∅)
87frgpgrp 18395 . . . . . . . 8 (∅ ∈ V → (freeGrp‘∅) ∈ Grp)
96, 8ax-mp 5 . . . . . . 7 (freeGrp‘∅) ∈ Grp
10 eqid 2760 . . . . . . . 8 (Base‘(freeGrp‘∅)) = (Base‘(freeGrp‘∅))
117, 100frgp 18412 . . . . . . 7 (Base‘(freeGrp‘∅)) ≈ 1𝑜
12100cyg 18514 . . . . . . 7 (((freeGrp‘∅) ∈ Grp ∧ (Base‘(freeGrp‘∅)) ≈ 1𝑜) → (freeGrp‘∅) ∈ CycGrp)
139, 11, 12mp2an 710 . . . . . 6 (freeGrp‘∅) ∈ CycGrp
145, 13syl6eqel 2847 . . . . 5 (𝐼 = ∅ → 𝐺 ∈ CycGrp)
152, 14sylbi 207 . . . 4 (𝐼 ≺ 1𝑜𝐺 ∈ CycGrp)
16 eqid 2760 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
17 eqid 2760 . . . . 5 (.g𝐺) = (.g𝐺)
18 relen 8128 . . . . . . 7 Rel ≈
1918brrelexi 5315 . . . . . 6 (𝐼 ≈ 1𝑜𝐼 ∈ V)
203frgpgrp 18395 . . . . . 6 (𝐼 ∈ V → 𝐺 ∈ Grp)
2119, 20syl 17 . . . . 5 (𝐼 ≈ 1𝑜𝐺 ∈ Grp)
22 eqid 2760 . . . . . . . 8 ( ~FG𝐼) = ( ~FG𝐼)
23 eqid 2760 . . . . . . . 8 (varFGrp𝐼) = (varFGrp𝐼)
2422, 23, 3, 16vrgpf 18401 . . . . . . 7 (𝐼 ∈ V → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
2519, 24syl 17 . . . . . 6 (𝐼 ≈ 1𝑜 → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
26 en1uniel 8195 . . . . . 6 (𝐼 ≈ 1𝑜 𝐼𝐼)
2725, 26ffvelrnd 6524 . . . . 5 (𝐼 ≈ 1𝑜 → ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺))
28 zringgrp 20045 . . . . . . . . 9 ring ∈ Grp
29 uniexg 7121 . . . . . . . . . . . 12 (𝐼 ∈ V → 𝐼 ∈ V)
3019, 29syl 17 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 𝐼 ∈ V)
31 1zzd 11620 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 → 1 ∈ ℤ)
3230, 31fsnd 6341 . . . . . . . . . 10 (𝐼 ≈ 1𝑜 → {⟨ 𝐼, 1⟩}:{ 𝐼}⟶ℤ)
33 en1b 8191 . . . . . . . . . . . 12 (𝐼 ≈ 1𝑜𝐼 = { 𝐼})
3433biimpi 206 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜𝐼 = { 𝐼})
3534feq2d 6192 . . . . . . . . . 10 (𝐼 ≈ 1𝑜 → ({⟨ 𝐼, 1⟩}:𝐼⟶ℤ ↔ {⟨ 𝐼, 1⟩}:{ 𝐼}⟶ℤ))
3632, 35mpbird 247 . . . . . . . . 9 (𝐼 ≈ 1𝑜 → {⟨ 𝐼, 1⟩}:𝐼⟶ℤ)
37 zringbas 20046 . . . . . . . . . 10 ℤ = (Base‘ℤring)
383, 37, 23frgpup3 18411 . . . . . . . . 9 ((ℤring ∈ Grp ∧ 𝐼 ∈ V ∧ {⟨ 𝐼, 1⟩}:𝐼⟶ℤ) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
3928, 19, 36, 38mp3an2i 1578 . . . . . . . 8 (𝐼 ≈ 1𝑜 → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
4039adantr 472 . . . . . . 7 ((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
41 reurex 3299 . . . . . . 7 (∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
4240, 41syl 17 . . . . . 6 ((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) → ∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
43 fveq1 6352 . . . . . . . . . 10 ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = ({⟨ 𝐼, 1⟩}‘ 𝐼))
44 fvco3 6438 . . . . . . . . . . . 12 (((varFGrp𝐼):𝐼⟶(Base‘𝐺) ∧ 𝐼𝐼) → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
4525, 26, 44syl2anc 696 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
46 1z 11619 . . . . . . . . . . . 12 1 ∈ ℤ
47 fvsng 6612 . . . . . . . . . . . 12 (( 𝐼 ∈ V ∧ 1 ∈ ℤ) → ({⟨ 𝐼, 1⟩}‘ 𝐼) = 1)
4830, 46, 47sylancl 697 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 → ({⟨ 𝐼, 1⟩}‘ 𝐼) = 1)
4945, 48eqeq12d 2775 . . . . . . . . . 10 (𝐼 ≈ 1𝑜 → (((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = ({⟨ 𝐼, 1⟩}‘ 𝐼) ↔ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
5043, 49syl5ib 234 . . . . . . . . 9 (𝐼 ≈ 1𝑜 → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
5150ad2antrr 764 . . . . . . . 8 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
5216, 37ghmf 17885 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐺 GrpHom ℤring) → 𝑓:(Base‘𝐺)⟶ℤ)
5352ad2antrl 766 . . . . . . . . . . . 12 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓:(Base‘𝐺)⟶ℤ)
5453ffvelrnda 6523 . . . . . . . . . . 11 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑥) ∈ ℤ)
5554an32s 881 . . . . . . . . . 10 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑓𝑥) ∈ ℤ)
56 mptresid 5614 . . . . . . . . . . . . . 14 (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = ( I ↾ (Base‘𝐺))
573, 16, 23frgpup3 18411 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝐼 ∈ V ∧ (varFGrp𝐼):𝐼⟶(Base‘𝐺)) → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
5821, 19, 25, 57syl3anc 1477 . . . . . . . . . . . . . . . . 17 (𝐼 ≈ 1𝑜 → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
59 reurmo 3300 . . . . . . . . . . . . . . . . 17 (∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
6058, 59syl 17 . . . . . . . . . . . . . . . 16 (𝐼 ≈ 1𝑜 → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
6160adantr 472 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
6221adantr 472 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝐺 ∈ Grp)
6316idghm 17896 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → ( I ↾ (Base‘𝐺)) ∈ (𝐺 GrpHom 𝐺))
6462, 63syl 17 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ( I ↾ (Base‘𝐺)) ∈ (𝐺 GrpHom 𝐺))
6525adantr 472 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
66 fcoi2 6240 . . . . . . . . . . . . . . . 16 ((varFGrp𝐼):𝐼⟶(Base‘𝐺) → (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)) = (varFGrp𝐼))
6765, 66syl 17 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)) = (varFGrp𝐼))
6853feqmptd 6412 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑓𝑥)))
69 eqidd 2761 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
70 oveq1 6821 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑓𝑥) → (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
7154, 68, 69, 70fmptco 6560 . . . . . . . . . . . . . . . 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))))
7227adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺))
73 eqid 2760 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
7417, 73, 16mulgghm2 20067 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺)) → (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (ℤring GrpHom 𝐺))
7562, 72, 74syl2anc 696 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (ℤring GrpHom 𝐺))
76 simprl 811 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓 ∈ (𝐺 GrpHom ℤring))
77 ghmco 17901 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (ℤring GrpHom 𝐺) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ 𝑓) ∈ (𝐺 GrpHom 𝐺))
7875, 76, 77syl2anc 696 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ 𝑓) ∈ (𝐺 GrpHom 𝐺))
7971, 78eqeltrrd 2840 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (𝐺 GrpHom 𝐺))
8034adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝐼 = { 𝐼})
8180eleq2d 2825 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼𝑦 ∈ { 𝐼}))
82 simprr 813 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)
8382oveq1d 6829 . . . . . . . . . . . . . . . . . . . . 21 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = (1(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
8416, 17mulg1 17769 . . . . . . . . . . . . . . . . . . . . . 22 (((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺) → (1(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼))
8572, 84syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (1(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼))
8683, 85eqtrd 2794 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼))
87 elsni 4338 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ { 𝐼} → 𝑦 = 𝐼)
8887fveq2d 6357 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ { 𝐼} → ((varFGrp𝐼)‘𝑦) = ((varFGrp𝐼)‘ 𝐼))
8988fveq2d 6357 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ { 𝐼} → (𝑓‘((varFGrp𝐼)‘𝑦)) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
9089oveq1d 6829 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ { 𝐼} → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
9190, 88eqeq12d 2775 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ { 𝐼} → (((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦) ↔ ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼)))
9286, 91syl5ibrcom 237 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦 ∈ { 𝐼} → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦)))
9381, 92sylbid 230 . . . . . . . . . . . . . . . . . 18 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼 → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦)))
9493imp 444 . . . . . . . . . . . . . . . . 17 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑦𝐼) → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦))
9594mpteq2dva 4896 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼 ↦ ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑦𝐼 ↦ ((varFGrp𝐼)‘𝑦)))
9665ffvelrnda 6523 . . . . . . . . . . . . . . . . 17 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑦𝐼) → ((varFGrp𝐼)‘𝑦) ∈ (Base‘𝐺))
9765feqmptd 6412 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (varFGrp𝐼) = (𝑦𝐼 ↦ ((varFGrp𝐼)‘𝑦)))
98 eqidd 2761 . . . . . . . . . . . . . . . . 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))))
99 fveq2 6353 . . . . . . . . . . . . . . . . . 18 (𝑥 = ((varFGrp𝐼)‘𝑦) → (𝑓𝑥) = (𝑓‘((varFGrp𝐼)‘𝑦)))
10099oveq1d 6829 . . . . . . . . . . . . . . . . 17 (𝑥 = ((varFGrp𝐼)‘𝑦) → ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
10196, 97, 98, 100fmptco 6560 . . . . . . . . . . . . . . . 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))))
10295, 101, 973eqtr4d 2804 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)) = (varFGrp𝐼))
103 coeq1 5435 . . . . . . . . . . . . . . . . 17 (𝑔 = ( I ↾ (Base‘𝐺)) → (𝑔 ∘ (varFGrp𝐼)) = (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)))
104103eqeq1d 2762 . . . . . . . . . . . . . . . 16 (𝑔 = ( I ↾ (Base‘𝐺)) → ((𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) ↔ (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)) = (varFGrp𝐼)))
105 coeq1 5435 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → (𝑔 ∘ (varFGrp𝐼)) = ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)))
106105eqeq1d 2762 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → ((𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)) = (varFGrp𝐼)))
107104, 106rmoi 3671 . . . . . . . . . . . . . . 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))))
10861, 64, 67, 79, 102, 107syl122anc 1486 . . . . . . . . . . . . . 14 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
10956, 108syl5eq 2806 . . . . . . . . . . . . 13 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
110 mpteqb 6462 . . . . . . . . . . . . . 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))))
111 id 22 . . . . . . . . . . . . . 14 (𝑥 ∈ (Base‘𝐺) → 𝑥 ∈ (Base‘𝐺))
112110, 111mprg 3064 . . . . . . . . . . . . 13 ((𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ↔ ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
113109, 112sylib 208 . . . . . . . . . . . 12 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
114113r19.21bi 3070 . . . . . . . . . . 11 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
115114an32s 881 . . . . . . . . . 10 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
11670eqeq2d 2770 . . . . . . . . . . 11 (𝑛 = (𝑓𝑥) → (𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)) ↔ 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
117116rspcev 3449 . . . . . . . . . 10 (((𝑓𝑥) ∈ ℤ ∧ 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
11855, 115, 117syl2anc 696 . . . . . . . . 9 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
119118expr 644 . . . . . . . 8 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1 → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
12051, 119syld 47 . . . . . . 7 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
121120rexlimdva 3169 . . . . . 6 ((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) → (∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
12242, 121mpd 15 . . . . 5 ((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
12316, 17, 21, 27, 122iscygd 18509 . . . 4 (𝐼 ≈ 1𝑜𝐺 ∈ CycGrp)
12415, 123jaoi 393 . . 3 ((𝐼 ≺ 1𝑜𝐼 ≈ 1𝑜) → 𝐺 ∈ CycGrp)
1251, 124sylbi 207 . 2 (𝐼 ≼ 1𝑜𝐺 ∈ CycGrp)
126 cygabl 18512 . . 3 (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
1273frgpnabl 18498 . . . . 5 (1𝑜𝐼 → ¬ 𝐺 ∈ Abel)
128127con2i 134 . . . 4 (𝐺 ∈ Abel → ¬ 1𝑜𝐼)
129 ablgrp 18418 . . . . . 6 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
130 eqid 2760 . . . . . . 7 (0g𝐺) = (0g𝐺)
13116, 130grpidcl 17671 . . . . . 6 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
1323, 16elbasfv 16142 . . . . . 6 ((0g𝐺) ∈ (Base‘𝐺) → 𝐼 ∈ V)
133129, 131, 1323syl 18 . . . . 5 (𝐺 ∈ Abel → 𝐼 ∈ V)
134 1onn 7890 . . . . . 6 1𝑜 ∈ ω
135 nnfi 8320 . . . . . 6 (1𝑜 ∈ ω → 1𝑜 ∈ Fin)
136134, 135ax-mp 5 . . . . 5 1𝑜 ∈ Fin
137 fidomtri2 9030 . . . . 5 ((𝐼 ∈ V ∧ 1𝑜 ∈ Fin) → (𝐼 ≼ 1𝑜 ↔ ¬ 1𝑜𝐼))
138133, 136, 137sylancl 697 . . . 4 (𝐺 ∈ Abel → (𝐼 ≼ 1𝑜 ↔ ¬ 1𝑜𝐼))
139128, 138mpbird 247 . . 3 (𝐺 ∈ Abel → 𝐼 ≼ 1𝑜)
140126, 139syl 17 . 2 (𝐺 ∈ CycGrp → 𝐼 ≼ 1𝑜)
141125, 140impbii 199 1 (𝐼 ≼ 1𝑜𝐺 ∈ CycGrp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  ∃!wreu 3052  ∃*wrmo 3053  Vcvv 3340  c0 4058  {csn 4321  cop 4327   cuni 4588   class class class wbr 4804  cmpt 4881   I cid 5173  cres 5268  ccom 5270  wf 6045  cfv 6049  (class class class)co 6814  ωcom 7231  1𝑜c1o 7723  cen 8120  cdom 8121  csdm 8122  Fincfn 8123  1c1 10149  cz 11589  Basecbs 16079  0gc0g 16322  Grpcgrp 17643  .gcmg 17761   GrpHom cghm 17878   ~FG cefg 18339  freeGrpcfrgp 18340  varFGrpcvrgp 18341  Abelcabl 18414  CycGrpccyg 18499  ringzring 20040
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  ax-addf 10227  ax-mulf 10228
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-ot 4330  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  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-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-ec 7915  df-qs 7919  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-inf 8516  df-card 8975  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-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-xnn0 11576  df-z 11590  df-dec 11706  df-uz 11900  df-rp 12046  df-fz 12540  df-fzo 12680  df-seq 13016  df-hash 13332  df-word 13505  df-lsw 13506  df-concat 13507  df-s1 13508  df-substr 13509  df-splice 13510  df-reverse 13511  df-s2 13813  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-ress 16087  df-plusg 16176  df-mulr 16177  df-starv 16178  df-sca 16179  df-vsca 16180  df-ip 16181  df-tset 16182  df-ple 16183  df-ds 16186  df-unif 16187  df-0g 16324  df-gsum 16325  df-imas 16390  df-qus 16391  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-mhm 17556  df-submnd 17557  df-frmd 17607  df-vrmd 17608  df-grp 17646  df-minusg 17647  df-mulg 17762  df-subg 17812  df-ghm 17879  df-efg 18342  df-frgp 18343  df-vrgp 18344  df-cmn 18415  df-abl 18416  df-cyg 18500  df-mgp 18710  df-ur 18722  df-ring 18769  df-cring 18770  df-subrg 19000  df-cnfld 19969  df-zring 20041
This theorem is referenced by: (None)
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