MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgpcyg Structured version   Visualization version   GIF version

Theorem frgpcyg 19682
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 7844 . . 3 (𝐼 ≼ 1𝑜 ↔ (𝐼 ≺ 1𝑜𝐼 ≈ 1𝑜))
2 sdom1 8018 . . . . 5 (𝐼 ≺ 1𝑜𝐼 = ∅)
3 frgpcyg.g . . . . . . 7 𝐺 = (freeGrp‘𝐼)
4 fveq2 6084 . . . . . . 7 (𝐼 = ∅ → (freeGrp‘𝐼) = (freeGrp‘∅))
53, 4syl5eq 2651 . . . . . 6 (𝐼 = ∅ → 𝐺 = (freeGrp‘∅))
6 0ex 4709 . . . . . . . 8 ∅ ∈ V
7 eqid 2605 . . . . . . . . 9 (freeGrp‘∅) = (freeGrp‘∅)
87frgpgrp 17940 . . . . . . . 8 (∅ ∈ V → (freeGrp‘∅) ∈ Grp)
96, 8ax-mp 5 . . . . . . 7 (freeGrp‘∅) ∈ Grp
10 eqid 2605 . . . . . . . 8 (Base‘(freeGrp‘∅)) = (Base‘(freeGrp‘∅))
117, 100frgp 17957 . . . . . . 7 (Base‘(freeGrp‘∅)) ≈ 1𝑜
12100cyg 18059 . . . . . . 7 (((freeGrp‘∅) ∈ Grp ∧ (Base‘(freeGrp‘∅)) ≈ 1𝑜) → (freeGrp‘∅) ∈ CycGrp)
139, 11, 12mp2an 703 . . . . . 6 (freeGrp‘∅) ∈ CycGrp
145, 13syl6eqel 2691 . . . . 5 (𝐼 = ∅ → 𝐺 ∈ CycGrp)
152, 14sylbi 205 . . . 4 (𝐼 ≺ 1𝑜𝐺 ∈ CycGrp)
16 eqid 2605 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
17 eqid 2605 . . . . 5 (.g𝐺) = (.g𝐺)
18 relen 7819 . . . . . . 7 Rel ≈
1918brrelexi 5068 . . . . . 6 (𝐼 ≈ 1𝑜𝐼 ∈ V)
203frgpgrp 17940 . . . . . 6 (𝐼 ∈ V → 𝐺 ∈ Grp)
2119, 20syl 17 . . . . 5 (𝐼 ≈ 1𝑜𝐺 ∈ Grp)
22 eqid 2605 . . . . . . . 8 ( ~FG𝐼) = ( ~FG𝐼)
23 eqid 2605 . . . . . . . 8 (varFGrp𝐼) = (varFGrp𝐼)
2422, 23, 3, 16vrgpf 17946 . . . . . . 7 (𝐼 ∈ V → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
2519, 24syl 17 . . . . . 6 (𝐼 ≈ 1𝑜 → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
26 en1uniel 7887 . . . . . 6 (𝐼 ≈ 1𝑜 𝐼𝐼)
2725, 26ffvelrnd 6249 . . . . 5 (𝐼 ≈ 1𝑜 → ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺))
28 zringgrp 19584 . . . . . . . . 9 ring ∈ Grp
29 uniexg 6826 . . . . . . . . . . . 12 (𝐼 ∈ V → 𝐼 ∈ V)
3019, 29syl 17 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 𝐼 ∈ V)
31 1zzd 11237 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 → 1 ∈ ℤ)
3230, 31fsnd 6072 . . . . . . . . . 10 (𝐼 ≈ 1𝑜 → {⟨ 𝐼, 1⟩}:{ 𝐼}⟶ℤ)
33 en1b 7883 . . . . . . . . . . . 12 (𝐼 ≈ 1𝑜𝐼 = { 𝐼})
3433biimpi 204 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜𝐼 = { 𝐼})
3534feq2d 5926 . . . . . . . . . 10 (𝐼 ≈ 1𝑜 → ({⟨ 𝐼, 1⟩}:𝐼⟶ℤ ↔ {⟨ 𝐼, 1⟩}:{ 𝐼}⟶ℤ))
3632, 35mpbird 245 . . . . . . . . 9 (𝐼 ≈ 1𝑜 → {⟨ 𝐼, 1⟩}:𝐼⟶ℤ)
37 zringbas 19585 . . . . . . . . . 10 ℤ = (Base‘ℤring)
383, 37, 23frgpup3 17956 . . . . . . . . 9 ((ℤring ∈ Grp ∧ 𝐼 ∈ V ∧ {⟨ 𝐼, 1⟩}:𝐼⟶ℤ) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
3928, 19, 36, 38mp3an2i 1420 . . . . . . . 8 (𝐼 ≈ 1𝑜 → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
4039adantr 479 . . . . . . 7 ((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
41 reurex 3132 . . . . . . 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 6083 . . . . . . . . . 10 ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = ({⟨ 𝐼, 1⟩}‘ 𝐼))
44 fvco3 6166 . . . . . . . . . . . 12 (((varFGrp𝐼):𝐼⟶(Base‘𝐺) ∧ 𝐼𝐼) → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
4525, 26, 44syl2anc 690 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
46 1z 11236 . . . . . . . . . . . 12 1 ∈ ℤ
47 fvsng 6326 . . . . . . . . . . . 12 (( 𝐼 ∈ V ∧ 1 ∈ ℤ) → ({⟨ 𝐼, 1⟩}‘ 𝐼) = 1)
4830, 46, 47sylancl 692 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 → ({⟨ 𝐼, 1⟩}‘ 𝐼) = 1)
4945, 48eqeq12d 2620 . . . . . . . . . 10 (𝐼 ≈ 1𝑜 → (((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = ({⟨ 𝐼, 1⟩}‘ 𝐼) ↔ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
5043, 49syl5ib 232 . . . . . . . . 9 (𝐼 ≈ 1𝑜 → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
5150ad2antrr 757 . . . . . . . 8 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
5216, 37ghmf 17429 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐺 GrpHom ℤring) → 𝑓:(Base‘𝐺)⟶ℤ)
5352ad2antrl 759 . . . . . . . . . . . 12 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓:(Base‘𝐺)⟶ℤ)
5453ffvelrnda 6248 . . . . . . . . . . 11 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑥) ∈ ℤ)
5554an32s 841 . . . . . . . . . 10 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑓𝑥) ∈ ℤ)
56 mptresid 5358 . . . . . . . . . . . . . 14 (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = ( I ↾ (Base‘𝐺))
573, 16, 23frgpup3 17956 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝐼 ∈ V ∧ (varFGrp𝐼):𝐼⟶(Base‘𝐺)) → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
5821, 19, 25, 57syl3anc 1317 . . . . . . . . . . . . . . . . 17 (𝐼 ≈ 1𝑜 → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
59 reurmo 3133 . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
6221adantr 479 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝐺 ∈ Grp)
6316idghm 17440 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
66 fcoi2 5973 . . . . . . . . . . . . . . . 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 6140 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑓𝑥)))
69 eqidd 2606 . . . . . . . . . . . . . . . . 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 6530 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑓𝑥) → (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
7154, 68, 69, 70fmptco 6284 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . 18 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺))
73 eqid 2605 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
7417, 73, 16mulgghm2 19605 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺)) → (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (ℤring GrpHom 𝐺))
7562, 72, 74syl2anc 690 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (ℤring GrpHom 𝐺))
76 simprl 789 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓 ∈ (𝐺 GrpHom ℤring))
77 ghmco 17445 . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ 𝑓) ∈ (𝐺 GrpHom 𝐺))
7971, 78eqeltrrd 2684 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (𝐺 GrpHom 𝐺))
8034adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝐼 = { 𝐼})
8180eleq2d 2668 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼𝑦 ∈ { 𝐼}))
82 simprr 791 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)
8382oveq1d 6538 . . . . . . . . . . . . . . . . . . . . 21 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = (1(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
8416, 17mulg1 17313 . . . . . . . . . . . . . . . . . . . . . 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 2639 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼))
87 elsni 4137 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ { 𝐼} → 𝑦 = 𝐼)
8887fveq2d 6088 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ { 𝐼} → ((varFGrp𝐼)‘𝑦) = ((varFGrp𝐼)‘ 𝐼))
8988fveq2d 6088 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ { 𝐼} → (𝑓‘((varFGrp𝐼)‘𝑦)) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
9089oveq1d 6538 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ { 𝐼} → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
9190, 88eqeq12d 2620 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ { 𝐼} → (((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦) ↔ ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼)))
9286, 91syl5ibrcom 235 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦 ∈ { 𝐼} → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦)))
9381, 92sylbid 228 . . . . . . . . . . . . . . . . . 18 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼 → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦)))
9493imp 443 . . . . . . . . . . . . . . . . 17 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑦𝐼) → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦))
9594mpteq2dva 4662 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼 ↦ ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑦𝐼 ↦ ((varFGrp𝐼)‘𝑦)))
9665ffvelrnda 6248 . . . . . . . . . . . . . . . . 17 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑦𝐼) → ((varFGrp𝐼)‘𝑦) ∈ (Base‘𝐺))
9765feqmptd 6140 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (varFGrp𝐼) = (𝑦𝐼 ↦ ((varFGrp𝐼)‘𝑦)))
98 eqidd 2606 . . . . . . . . . . . . . . . . 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 6084 . . . . . . . . . . . . . . . . . 18 (𝑥 = ((varFGrp𝐼)‘𝑦) → (𝑓𝑥) = (𝑓‘((varFGrp𝐼)‘𝑦)))
10099oveq1d 6538 . . . . . . . . . . . . . . . . 17 (𝑥 = ((varFGrp𝐼)‘𝑦) → ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
10196, 97, 98, 100fmptco 6284 . . . . . . . . . . . . . . . 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 2649 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)) = (varFGrp𝐼))
103 coeq1 5185 . . . . . . . . . . . . . . . . 17 (𝑔 = ( I ↾ (Base‘𝐺)) → (𝑔 ∘ (varFGrp𝐼)) = (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)))
104103eqeq1d 2607 . . . . . . . . . . . . . . . 16 (𝑔 = ( I ↾ (Base‘𝐺)) → ((𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) ↔ (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)) = (varFGrp𝐼)))
105 coeq1 5185 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → (𝑔 ∘ (varFGrp𝐼)) = ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)))
106105eqeq1d 2607 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → ((𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)) = (varFGrp𝐼)))
107104, 106rmoi 3491 . . . . . . . . . . . . . . 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 1326 . . . . . . . . . . . . . 14 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
10956, 108syl5eq 2651 . . . . . . . . . . . . 13 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
110 mpteqb 6188 . . . . . . . . . . . . . 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 2905 . . . . . . . . . . . . 13 ((𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ↔ ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
113109, 112sylib 206 . . . . . . . . . . . 12 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∀𝑥 ∈ (Base‘𝐺)𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
114113r19.21bi 2911 . . . . . . . . . . 11 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
115114an32s 841 . . . . . . . . . 10 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
11670eqeq2d 2615 . . . . . . . . . . 11 (𝑛 = (𝑓𝑥) → (𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)) ↔ 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
117116rspcev 3277 . . . . . . . . . 10 (((𝑓𝑥) ∈ ℤ ∧ 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
11855, 115, 117syl2anc 690 . . . . . . . . 9 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
119118expr 640 . . . . . . . 8 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1 → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
12051, 119syld 45 . . . . . . 7 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
121120rexlimdva 3008 . . . . . 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 18054 . . . 4 (𝐼 ≈ 1𝑜𝐺 ∈ CycGrp)
12415, 123jaoi 392 . . 3 ((𝐼 ≺ 1𝑜𝐼 ≈ 1𝑜) → 𝐺 ∈ CycGrp)
1251, 124sylbi 205 . 2 (𝐼 ≼ 1𝑜𝐺 ∈ CycGrp)
126 cygabl 18057 . . 3 (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
1273frgpnabl 18043 . . . . 5 (1𝑜𝐼 → ¬ 𝐺 ∈ Abel)
128127con2i 132 . . . 4 (𝐺 ∈ Abel → ¬ 1𝑜𝐼)
129 ablgrp 17963 . . . . . 6 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
130 eqid 2605 . . . . . . 7 (0g𝐺) = (0g𝐺)
13116, 130grpidcl 17215 . . . . . 6 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
1323, 16elbasfv 15690 . . . . . 6 ((0g𝐺) ∈ (Base‘𝐺) → 𝐼 ∈ V)
133129, 131, 1323syl 18 . . . . 5 (𝐺 ∈ Abel → 𝐼 ∈ V)
134 1onn 7579 . . . . . 6 1𝑜 ∈ ω
135 nnfi 8011 . . . . . 6 (1𝑜 ∈ ω → 1𝑜 ∈ Fin)
136134, 135ax-mp 5 . . . . 5 1𝑜 ∈ Fin
137 fidomtri2 8676 . . . . 5 ((𝐼 ∈ V ∧ 1𝑜 ∈ Fin) → (𝐼 ≼ 1𝑜 ↔ ¬ 1𝑜𝐼))
138133, 136, 137sylancl 692 . . . 4 (𝐺 ∈ Abel → (𝐼 ≼ 1𝑜 ↔ ¬ 1𝑜𝐼))
139128, 138mpbird 245 . . 3 (𝐺 ∈ Abel → 𝐼 ≼ 1𝑜)
140126, 139syl 17 . 2 (𝐺 ∈ CycGrp → 𝐼 ≼ 1𝑜)
141125, 140impbii 197 1 (𝐼 ≼ 1𝑜𝐺 ∈ CycGrp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1975  wral 2891  wrex 2892  ∃!wreu 2893  ∃*wrmo 2894  Vcvv 3168  c0 3869  {csn 4120  cop 4126   cuni 4362   class class class wbr 4573  cmpt 4633   I cid 4934  cres 5026  ccom 5028  wf 5782  cfv 5786  (class class class)co 6523  ωcom 6930  1𝑜c1o 7413  cen 7811  cdom 7812  csdm 7813  Fincfn 7814  1c1 9789  cz 11206  Basecbs 15637  0gc0g 15865  Grpcgrp 17187  .gcmg 17305   GrpHom cghm 17422   ~FG cefg 17884  freeGrpcfrgp 17885  varFGrpcvrgp 17886  Abelcabl 17959  CycGrpccyg 18044  ringzring 19579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-addf 9867  ax-mulf 9868
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-ot 4129  df-uni 4363  df-int 4401  df-iun 4447  df-iin 4448  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-ec 7604  df-qs 7608  df-map 7719  df-pm 7720  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-sup 8204  df-inf 8205  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-3 10923  df-4 10924  df-5 10925  df-6 10926  df-7 10927  df-8 10928  df-9 10929  df-n0 11136  df-z 11207  df-dec 11322  df-uz 11516  df-rp 11661  df-fz 12149  df-fzo 12286  df-seq 12615  df-hash 12931  df-word 13096  df-lsw 13097  df-concat 13098  df-s1 13099  df-substr 13100  df-splice 13101  df-reverse 13102  df-s2 13386  df-struct 15639  df-ndx 15640  df-slot 15641  df-base 15642  df-sets 15643  df-ress 15644  df-plusg 15723  df-mulr 15724  df-starv 15725  df-sca 15726  df-vsca 15727  df-ip 15728  df-tset 15729  df-ple 15730  df-ds 15733  df-unif 15734  df-0g 15867  df-gsum 15868  df-imas 15933  df-qus 15934  df-mgm 17007  df-sgrp 17049  df-mnd 17060  df-mhm 17100  df-submnd 17101  df-frmd 17151  df-vrmd 17152  df-grp 17190  df-minusg 17191  df-mulg 17306  df-subg 17356  df-ghm 17423  df-efg 17887  df-frgp 17888  df-vrgp 17889  df-cmn 17960  df-abl 17961  df-cyg 18045  df-mgp 18255  df-ur 18267  df-ring 18314  df-cring 18315  df-subrg 18543  df-cnfld 19510  df-zring 19580
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator