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Theorem frgpcyg 20129
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 8222 . . 3 (𝐼 ≼ 1𝑜 ↔ (𝐼 ≺ 1𝑜𝐼 ≈ 1𝑜))
2 sdom1 8399 . . . . 5 (𝐼 ≺ 1𝑜𝐼 = ∅)
3 frgpcyg.g . . . . . . 7 𝐺 = (freeGrp‘𝐼)
4 fveq2 6408 . . . . . . 7 (𝐼 = ∅ → (freeGrp‘𝐼) = (freeGrp‘∅))
53, 4syl5eq 2852 . . . . . 6 (𝐼 = ∅ → 𝐺 = (freeGrp‘∅))
6 0ex 4984 . . . . . . . 8 ∅ ∈ V
7 eqid 2806 . . . . . . . . 9 (freeGrp‘∅) = (freeGrp‘∅)
87frgpgrp 18376 . . . . . . . 8 (∅ ∈ V → (freeGrp‘∅) ∈ Grp)
96, 8ax-mp 5 . . . . . . 7 (freeGrp‘∅) ∈ Grp
10 eqid 2806 . . . . . . . 8 (Base‘(freeGrp‘∅)) = (Base‘(freeGrp‘∅))
117, 100frgp 18393 . . . . . . 7 (Base‘(freeGrp‘∅)) ≈ 1𝑜
12100cyg 18495 . . . . . . 7 (((freeGrp‘∅) ∈ Grp ∧ (Base‘(freeGrp‘∅)) ≈ 1𝑜) → (freeGrp‘∅) ∈ CycGrp)
139, 11, 12mp2an 675 . . . . . 6 (freeGrp‘∅) ∈ CycGrp
145, 13syl6eqel 2893 . . . . 5 (𝐼 = ∅ → 𝐺 ∈ CycGrp)
152, 14sylbi 208 . . . 4 (𝐼 ≺ 1𝑜𝐺 ∈ CycGrp)
16 eqid 2806 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
17 eqid 2806 . . . . 5 (.g𝐺) = (.g𝐺)
18 relen 8197 . . . . . . 7 Rel ≈
1918brrelexi 5358 . . . . . 6 (𝐼 ≈ 1𝑜𝐼 ∈ V)
203frgpgrp 18376 . . . . . 6 (𝐼 ∈ V → 𝐺 ∈ Grp)
2119, 20syl 17 . . . . 5 (𝐼 ≈ 1𝑜𝐺 ∈ Grp)
22 eqid 2806 . . . . . . . 8 ( ~FG𝐼) = ( ~FG𝐼)
23 eqid 2806 . . . . . . . 8 (varFGrp𝐼) = (varFGrp𝐼)
2422, 23, 3, 16vrgpf 18382 . . . . . . 7 (𝐼 ∈ V → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
2519, 24syl 17 . . . . . 6 (𝐼 ≈ 1𝑜 → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
26 en1uniel 8264 . . . . . 6 (𝐼 ≈ 1𝑜 𝐼𝐼)
2725, 26ffvelrnd 6582 . . . . 5 (𝐼 ≈ 1𝑜 → ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺))
28 zringgrp 20031 . . . . . . . . 9 ring ∈ Grp
29 uniexg 7185 . . . . . . . . . . . 12 (𝐼 ∈ V → 𝐼 ∈ V)
3019, 29syl 17 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 𝐼 ∈ V)
31 1zzd 11674 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 → 1 ∈ ℤ)
3230, 31fsnd 6395 . . . . . . . . . 10 (𝐼 ≈ 1𝑜 → {⟨ 𝐼, 1⟩}:{ 𝐼}⟶ℤ)
33 en1b 8260 . . . . . . . . . . . 12 (𝐼 ≈ 1𝑜𝐼 = { 𝐼})
3433biimpi 207 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜𝐼 = { 𝐼})
3534feq2d 6242 . . . . . . . . . 10 (𝐼 ≈ 1𝑜 → ({⟨ 𝐼, 1⟩}:𝐼⟶ℤ ↔ {⟨ 𝐼, 1⟩}:{ 𝐼}⟶ℤ))
3632, 35mpbird 248 . . . . . . . . 9 (𝐼 ≈ 1𝑜 → {⟨ 𝐼, 1⟩}:𝐼⟶ℤ)
37 zringbas 20032 . . . . . . . . . 10 ℤ = (Base‘ℤring)
383, 37, 23frgpup3 18392 . . . . . . . . 9 ((ℤring ∈ Grp ∧ 𝐼 ∈ V ∧ {⟨ 𝐼, 1⟩}:𝐼⟶ℤ) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
3928, 19, 36, 38mp3an2i 1583 . . . . . . . 8 (𝐼 ≈ 1𝑜 → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
4039adantr 468 . . . . . . 7 ((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
41 reurex 3349 . . . . . . 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 6407 . . . . . . . . . 10 ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = ({⟨ 𝐼, 1⟩}‘ 𝐼))
44 fvco3 6496 . . . . . . . . . . . 12 (((varFGrp𝐼):𝐼⟶(Base‘𝐺) ∧ 𝐼𝐼) → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
4525, 26, 44syl2anc 575 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
46 1z 11673 . . . . . . . . . . . 12 1 ∈ ℤ
47 fvsng 6672 . . . . . . . . . . . 12 (( 𝐼 ∈ V ∧ 1 ∈ ℤ) → ({⟨ 𝐼, 1⟩}‘ 𝐼) = 1)
4830, 46, 47sylancl 576 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 → ({⟨ 𝐼, 1⟩}‘ 𝐼) = 1)
4945, 48eqeq12d 2821 . . . . . . . . . 10 (𝐼 ≈ 1𝑜 → (((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = ({⟨ 𝐼, 1⟩}‘ 𝐼) ↔ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
5043, 49syl5ib 235 . . . . . . . . 9 (𝐼 ≈ 1𝑜 → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
5150ad2antrr 708 . . . . . . . 8 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
5216, 37ghmf 17866 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐺 GrpHom ℤring) → 𝑓:(Base‘𝐺)⟶ℤ)
5352ad2antrl 710 . . . . . . . . . . . 12 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓:(Base‘𝐺)⟶ℤ)
5453ffvelrnda 6581 . . . . . . . . . . 11 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑥) ∈ ℤ)
5554an32s 634 . . . . . . . . . 10 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑓𝑥) ∈ ℤ)
56 mptresid 5668 . . . . . . . . . . . . . 14 (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = ( I ↾ (Base‘𝐺))
573, 16, 23frgpup3 18392 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝐼 ∈ V ∧ (varFGrp𝐼):𝐼⟶(Base‘𝐺)) → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
5821, 19, 25, 57syl3anc 1483 . . . . . . . . . . . . . . . . 17 (𝐼 ≈ 1𝑜 → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
59 reurmo 3350 . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
6221adantr 468 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝐺 ∈ Grp)
6316idghm 17877 . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
66 fcoi2 6294 . . . . . . . . . . . . . . . 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 6470 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑓𝑥)))
69 eqidd 2807 . . . . . . . . . . . . . . . . 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 6881 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑓𝑥) → (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
7154, 68, 69, 70fmptco 6619 . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . 18 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺))
73 eqid 2806 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
7417, 73, 16mulgghm2 20053 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺)) → (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (ℤring GrpHom 𝐺))
7562, 72, 74syl2anc 575 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (ℤring GrpHom 𝐺))
76 simprl 778 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓 ∈ (𝐺 GrpHom ℤring))
77 ghmco 17882 . . . . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ 𝑓) ∈ (𝐺 GrpHom 𝐺))
7971, 78eqeltrrd 2886 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (𝐺 GrpHom 𝐺))
8034adantr 468 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝐼 = { 𝐼})
8180eleq2d 2871 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼𝑦 ∈ { 𝐼}))
82 simprr 780 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)
8382oveq1d 6889 . . . . . . . . . . . . . . . . . . . . 21 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = (1(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
8416, 17mulg1 17753 . . . . . . . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼))
87 elsni 4387 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ { 𝐼} → 𝑦 = 𝐼)
8887fveq2d 6412 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ { 𝐼} → ((varFGrp𝐼)‘𝑦) = ((varFGrp𝐼)‘ 𝐼))
8988fveq2d 6412 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ { 𝐼} → (𝑓‘((varFGrp𝐼)‘𝑦)) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
9089oveq1d 6889 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ { 𝐼} → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
9190, 88eqeq12d 2821 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ { 𝐼} → (((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦) ↔ ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼)))
9286, 91syl5ibrcom 238 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦 ∈ { 𝐼} → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦)))
9381, 92sylbid 231 . . . . . . . . . . . . . . . . . 18 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼 → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦)))
9493imp 395 . . . . . . . . . . . . . . . . 17 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑦𝐼) → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦))
9594mpteq2dva 4938 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼 ↦ ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑦𝐼 ↦ ((varFGrp𝐼)‘𝑦)))
9665ffvelrnda 6581 . . . . . . . . . . . . . . . . 17 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑦𝐼) → ((varFGrp𝐼)‘𝑦) ∈ (Base‘𝐺))
9765feqmptd 6470 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (varFGrp𝐼) = (𝑦𝐼 ↦ ((varFGrp𝐼)‘𝑦)))
98 eqidd 2807 . . . . . . . . . . . . . . . . 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 6408 . . . . . . . . . . . . . . . . . 18 (𝑥 = ((varFGrp𝐼)‘𝑦) → (𝑓𝑥) = (𝑓‘((varFGrp𝐼)‘𝑦)))
10099oveq1d 6889 . . . . . . . . . . . . . . . . 17 (𝑥 = ((varFGrp𝐼)‘𝑦) → ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
10196, 97, 98, 100fmptco 6619 . . . . . . . . . . . . . . . 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 2850 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)) = (varFGrp𝐼))
103 coeq1 5481 . . . . . . . . . . . . . . . . 17 (𝑔 = ( I ↾ (Base‘𝐺)) → (𝑔 ∘ (varFGrp𝐼)) = (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)))
104103eqeq1d 2808 . . . . . . . . . . . . . . . 16 (𝑔 = ( I ↾ (Base‘𝐺)) → ((𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) ↔ (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)) = (varFGrp𝐼)))
105 coeq1 5481 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → (𝑔 ∘ (varFGrp𝐼)) = ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)))
106105eqeq1d 2808 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → ((𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)) = (varFGrp𝐼)))
107104, 106rmoi 3725 . . . . . . . . . . . . . . 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 1491 . . . . . . . . . . . . . 14 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
10956, 108syl5eq 2852 . . . . . . . . . . . . 13 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
110 mpteqb 6520 . . . . . . . . . . . . . 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 3114 . . . . . . . . . . . . 13 ((𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ↔ ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
113109, 112sylib 209 . . . . . . . . . . . 12 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
114113r19.21bi 3120 . . . . . . . . . . 11 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
115114an32s 634 . . . . . . . . . 10 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
11670rspceeqv 3520 . . . . . . . . . 10 (((𝑓𝑥) ∈ ℤ ∧ 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
11755, 115, 116syl2anc 575 . . . . . . . . 9 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
118117expr 446 . . . . . . . 8 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1 → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
11951, 118syld 47 . . . . . . 7 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
120119rexlimdva 3219 . . . . . 6 ((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) → (∃𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
12142, 120mpd 15 . . . . 5 ((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
12216, 17, 21, 27, 121iscygd 18490 . . . 4 (𝐼 ≈ 1𝑜𝐺 ∈ CycGrp)
12315, 122jaoi 875 . . 3 ((𝐼 ≺ 1𝑜𝐼 ≈ 1𝑜) → 𝐺 ∈ CycGrp)
1241, 123sylbi 208 . 2 (𝐼 ≼ 1𝑜𝐺 ∈ CycGrp)
125 cygabl 18493 . . 3 (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
1263frgpnabl 18479 . . . . 5 (1𝑜𝐼 → ¬ 𝐺 ∈ Abel)
127126con2i 136 . . . 4 (𝐺 ∈ Abel → ¬ 1𝑜𝐼)
128 ablgrp 18399 . . . . . 6 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
129 eqid 2806 . . . . . . 7 (0g𝐺) = (0g𝐺)
13016, 129grpidcl 17655 . . . . . 6 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
1313, 16elbasfv 16131 . . . . . 6 ((0g𝐺) ∈ (Base‘𝐺) → 𝐼 ∈ V)
132128, 130, 1313syl 18 . . . . 5 (𝐺 ∈ Abel → 𝐼 ∈ V)
133 1onn 7956 . . . . . 6 1𝑜 ∈ ω
134 nnfi 8392 . . . . . 6 (1𝑜 ∈ ω → 1𝑜 ∈ Fin)
135133, 134ax-mp 5 . . . . 5 1𝑜 ∈ Fin
136 fidomtri2 9103 . . . . 5 ((𝐼 ∈ V ∧ 1𝑜 ∈ Fin) → (𝐼 ≼ 1𝑜 ↔ ¬ 1𝑜𝐼))
137132, 135, 136sylancl 576 . . . 4 (𝐺 ∈ Abel → (𝐼 ≼ 1𝑜 ↔ ¬ 1𝑜𝐼))
138127, 137mpbird 248 . . 3 (𝐺 ∈ Abel → 𝐼 ≼ 1𝑜)
139125, 138syl 17 . 2 (𝐺 ∈ CycGrp → 𝐼 ≼ 1𝑜)
140124, 139impbii 200 1 (𝐼 ≼ 1𝑜𝐺 ∈ CycGrp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wcel 2156  wral 3096  wrex 3097  ∃!wreu 3098  ∃*wrmo 3099  Vcvv 3391  c0 4116  {csn 4370  cop 4376   cuni 4630   class class class wbr 4844  cmpt 4923   I cid 5218  cres 5313  ccom 5315  wf 6097  cfv 6101  (class class class)co 6874  ωcom 7295  1𝑜c1o 7789  cen 8189  cdom 8190  csdm 8191  Fincfn 8192  1c1 10222  cz 11643  Basecbs 16068  0gc0g 16305  Grpcgrp 17627  .gcmg 17745   GrpHom cghm 17859   ~FG cefg 18320  freeGrpcfrgp 18321  varFGrpcvrgp 18322  Abelcabl 18395  CycGrpccyg 18480  ringzring 20026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298  ax-addf 10300  ax-mulf 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-ot 4379  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-er 7979  df-ec 7981  df-qs 7985  df-map 8094  df-pm 8095  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-sup 8587  df-inf 8588  df-card 9048  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-nn 11306  df-2 11364  df-3 11365  df-4 11366  df-5 11367  df-6 11368  df-7 11369  df-8 11370  df-9 11371  df-n0 11560  df-xnn0 11630  df-z 11644  df-dec 11760  df-uz 11905  df-rp 12047  df-fz 12550  df-fzo 12690  df-seq 13025  df-hash 13338  df-word 13510  df-lsw 13511  df-concat 13512  df-s1 13513  df-substr 13514  df-splice 13515  df-reverse 13516  df-s2 13817  df-struct 16070  df-ndx 16071  df-slot 16072  df-base 16074  df-sets 16075  df-ress 16076  df-plusg 16166  df-mulr 16167  df-starv 16168  df-sca 16169  df-vsca 16170  df-ip 16171  df-tset 16172  df-ple 16173  df-ds 16175  df-unif 16176  df-0g 16307  df-gsum 16308  df-imas 16373  df-qus 16374  df-mgm 17447  df-sgrp 17489  df-mnd 17500  df-mhm 17540  df-submnd 17541  df-frmd 17591  df-vrmd 17592  df-grp 17630  df-minusg 17631  df-mulg 17746  df-subg 17793  df-ghm 17860  df-efg 18323  df-frgp 18324  df-vrgp 18325  df-cmn 18396  df-abl 18397  df-cyg 18481  df-mgp 18692  df-ur 18704  df-ring 18751  df-cring 18752  df-subrg 18982  df-cnfld 19955  df-zring 20027
This theorem is referenced by: (None)
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