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Theorem frgpcyg 19916
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 7982 . . 3 (𝐼 ≼ 1𝑜 ↔ (𝐼 ≺ 1𝑜𝐼 ≈ 1𝑜))
2 sdom1 8157 . . . . 5 (𝐼 ≺ 1𝑜𝐼 = ∅)
3 frgpcyg.g . . . . . . 7 𝐺 = (freeGrp‘𝐼)
4 fveq2 6189 . . . . . . 7 (𝐼 = ∅ → (freeGrp‘𝐼) = (freeGrp‘∅))
53, 4syl5eq 2667 . . . . . 6 (𝐼 = ∅ → 𝐺 = (freeGrp‘∅))
6 0ex 4788 . . . . . . . 8 ∅ ∈ V
7 eqid 2621 . . . . . . . . 9 (freeGrp‘∅) = (freeGrp‘∅)
87frgpgrp 18169 . . . . . . . 8 (∅ ∈ V → (freeGrp‘∅) ∈ Grp)
96, 8ax-mp 5 . . . . . . 7 (freeGrp‘∅) ∈ Grp
10 eqid 2621 . . . . . . . 8 (Base‘(freeGrp‘∅)) = (Base‘(freeGrp‘∅))
117, 100frgp 18186 . . . . . . 7 (Base‘(freeGrp‘∅)) ≈ 1𝑜
12100cyg 18288 . . . . . . 7 (((freeGrp‘∅) ∈ Grp ∧ (Base‘(freeGrp‘∅)) ≈ 1𝑜) → (freeGrp‘∅) ∈ CycGrp)
139, 11, 12mp2an 708 . . . . . 6 (freeGrp‘∅) ∈ CycGrp
145, 13syl6eqel 2708 . . . . 5 (𝐼 = ∅ → 𝐺 ∈ CycGrp)
152, 14sylbi 207 . . . 4 (𝐼 ≺ 1𝑜𝐺 ∈ CycGrp)
16 eqid 2621 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
17 eqid 2621 . . . . 5 (.g𝐺) = (.g𝐺)
18 relen 7957 . . . . . . 7 Rel ≈
1918brrelexi 5156 . . . . . 6 (𝐼 ≈ 1𝑜𝐼 ∈ V)
203frgpgrp 18169 . . . . . 6 (𝐼 ∈ V → 𝐺 ∈ Grp)
2119, 20syl 17 . . . . 5 (𝐼 ≈ 1𝑜𝐺 ∈ Grp)
22 eqid 2621 . . . . . . . 8 ( ~FG𝐼) = ( ~FG𝐼)
23 eqid 2621 . . . . . . . 8 (varFGrp𝐼) = (varFGrp𝐼)
2422, 23, 3, 16vrgpf 18175 . . . . . . 7 (𝐼 ∈ V → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
2519, 24syl 17 . . . . . 6 (𝐼 ≈ 1𝑜 → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
26 en1uniel 8025 . . . . . 6 (𝐼 ≈ 1𝑜 𝐼𝐼)
2725, 26ffvelrnd 6358 . . . . 5 (𝐼 ≈ 1𝑜 → ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺))
28 zringgrp 19817 . . . . . . . . 9 ring ∈ Grp
29 uniexg 6952 . . . . . . . . . . . 12 (𝐼 ∈ V → 𝐼 ∈ V)
3019, 29syl 17 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 𝐼 ∈ V)
31 1zzd 11405 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 → 1 ∈ ℤ)
3230, 31fsnd 6177 . . . . . . . . . 10 (𝐼 ≈ 1𝑜 → {⟨ 𝐼, 1⟩}:{ 𝐼}⟶ℤ)
33 en1b 8021 . . . . . . . . . . . 12 (𝐼 ≈ 1𝑜𝐼 = { 𝐼})
3433biimpi 206 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜𝐼 = { 𝐼})
3534feq2d 6029 . . . . . . . . . 10 (𝐼 ≈ 1𝑜 → ({⟨ 𝐼, 1⟩}:𝐼⟶ℤ ↔ {⟨ 𝐼, 1⟩}:{ 𝐼}⟶ℤ))
3632, 35mpbird 247 . . . . . . . . 9 (𝐼 ≈ 1𝑜 → {⟨ 𝐼, 1⟩}:𝐼⟶ℤ)
37 zringbas 19818 . . . . . . . . . 10 ℤ = (Base‘ℤring)
383, 37, 23frgpup3 18185 . . . . . . . . 9 ((ℤring ∈ Grp ∧ 𝐼 ∈ V ∧ {⟨ 𝐼, 1⟩}:𝐼⟶ℤ) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
3928, 19, 36, 38mp3an2i 1428 . . . . . . . 8 (𝐼 ≈ 1𝑜 → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
4039adantr 481 . . . . . . 7 ((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) → ∃!𝑓 ∈ (𝐺 GrpHom ℤring)(𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩})
41 reurex 3158 . . . . . . 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 6188 . . . . . . . . . 10 ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = ({⟨ 𝐼, 1⟩}‘ 𝐼))
44 fvco3 6273 . . . . . . . . . . . 12 (((varFGrp𝐼):𝐼⟶(Base‘𝐺) ∧ 𝐼𝐼) → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
4525, 26, 44syl2anc 693 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 → ((𝑓 ∘ (varFGrp𝐼))‘ 𝐼) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
46 1z 11404 . . . . . . . . . . . 12 1 ∈ ℤ
47 fvsng 6444 . . . . . . . . . . . 12 (( 𝐼 ∈ V ∧ 1 ∈ ℤ) → ({⟨ 𝐼, 1⟩}‘ 𝐼) = 1)
4830, 46, 47sylancl 694 . . . . . . . . . . 11 (𝐼 ≈ 1𝑜 → ({⟨ 𝐼, 1⟩}‘ 𝐼) = 1)
4945, 48eqeq12d 2636 . . . . . . . . . 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 762 . . . . . . . 8 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ 𝑓 ∈ (𝐺 GrpHom ℤring)) → ((𝑓 ∘ (varFGrp𝐼)) = {⟨ 𝐼, 1⟩} → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1))
5216, 37ghmf 17658 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐺 GrpHom ℤring) → 𝑓:(Base‘𝐺)⟶ℤ)
5352ad2antrl 764 . . . . . . . . . . . 12 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓:(Base‘𝐺)⟶ℤ)
5453ffvelrnda 6357 . . . . . . . . . . 11 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑓𝑥) ∈ ℤ)
5554an32s 846 . . . . . . . . . 10 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑓𝑥) ∈ ℤ)
56 mptresid 5454 . . . . . . . . . . . . . 14 (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = ( I ↾ (Base‘𝐺))
573, 16, 23frgpup3 18185 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝐼 ∈ V ∧ (varFGrp𝐼):𝐼⟶(Base‘𝐺)) → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
5821, 19, 25, 57syl3anc 1325 . . . . . . . . . . . . . . . . 17 (𝐼 ≈ 1𝑜 → ∃!𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
59 reurmo 3159 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∃*𝑔 ∈ (𝐺 GrpHom 𝐺)(𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼))
6221adantr 481 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝐺 ∈ Grp)
6316idghm 17669 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (varFGrp𝐼):𝐼⟶(Base‘𝐺))
66 fcoi2 6077 . . . . . . . . . . . . . . . 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 6247 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑓𝑥)))
69 eqidd 2622 . . . . . . . . . . . . . . . . 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 6654 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑓𝑥) → (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
7154, 68, 69, 70fmptco 6394 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 18 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺))
73 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
7417, 73, 16mulgghm2 19839 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ ((varFGrp𝐼)‘ 𝐼) ∈ (Base‘𝐺)) → (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (ℤring GrpHom 𝐺))
7562, 72, 74syl2anc 693 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (ℤring GrpHom 𝐺))
76 simprl 794 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑓 ∈ (𝐺 GrpHom ℤring))
77 ghmco 17674 . . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑛 ∈ ℤ ↦ (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ 𝑓) ∈ (𝐺 GrpHom 𝐺))
7971, 78eqeltrrd 2701 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∈ (𝐺 GrpHom 𝐺))
8034adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝐼 = { 𝐼})
8180eleq2d 2686 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼𝑦 ∈ { 𝐼}))
82 simprr 796 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)
8382oveq1d 6662 . . . . . . . . . . . . . . . . . . . . 21 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = (1(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
8416, 17mulg1 17542 . . . . . . . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . . . . . . . 20 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘ 𝐼))
87 elsni 4192 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ { 𝐼} → 𝑦 = 𝐼)
8887fveq2d 6193 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ { 𝐼} → ((varFGrp𝐼)‘𝑦) = ((varFGrp𝐼)‘ 𝐼))
8988fveq2d 6193 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ { 𝐼} → (𝑓‘((varFGrp𝐼)‘𝑦)) = (𝑓‘((varFGrp𝐼)‘ 𝐼)))
9089oveq1d 6662 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ { 𝐼} → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓‘((varFGrp𝐼)‘ 𝐼))(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
9190, 88eqeq12d 2636 . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . 17 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑦𝐼) → ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((varFGrp𝐼)‘𝑦))
9594mpteq2dva 4742 . . . . . . . . . . . . . . . 16 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑦𝐼 ↦ ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼))) = (𝑦𝐼 ↦ ((varFGrp𝐼)‘𝑦)))
9665ffvelrnda 6357 . . . . . . . . . . . . . . . . 17 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑦𝐼) → ((varFGrp𝐼)‘𝑦) ∈ (Base‘𝐺))
9765feqmptd 6247 . . . . . . . . . . . . . . . . 17 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (varFGrp𝐼) = (𝑦𝐼 ↦ ((varFGrp𝐼)‘𝑦)))
98 eqidd 2622 . . . . . . . . . . . . . . . . 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 6189 . . . . . . . . . . . . . . . . . 18 (𝑥 = ((varFGrp𝐼)‘𝑦) → (𝑓𝑥) = (𝑓‘((varFGrp𝐼)‘𝑦)))
10099oveq1d 6662 . . . . . . . . . . . . . . . . 17 (𝑥 = ((varFGrp𝐼)‘𝑦) → ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)) = ((𝑓‘((varFGrp𝐼)‘𝑦))(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
10196, 97, 98, 100fmptco 6394 . . . . . . . . . . . . . . . 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 2665 . . . . . . . . . . . . . . 15 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)) = (varFGrp𝐼))
103 coeq1 5277 . . . . . . . . . . . . . . . . 17 (𝑔 = ( I ↾ (Base‘𝐺)) → (𝑔 ∘ (varFGrp𝐼)) = (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)))
104103eqeq1d 2623 . . . . . . . . . . . . . . . 16 (𝑔 = ( I ↾ (Base‘𝐺)) → ((𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) ↔ (( I ↾ (Base‘𝐺)) ∘ (varFGrp𝐼)) = (varFGrp𝐼)))
105 coeq1 5277 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → (𝑔 ∘ (varFGrp𝐼)) = ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)))
106105eqeq1d 2623 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → ((𝑔 ∘ (varFGrp𝐼)) = (varFGrp𝐼) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) ∘ (varFGrp𝐼)) = (varFGrp𝐼)))
107104, 106rmoi 3528 . . . . . . . . . . . . . . 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 1334 . . . . . . . . . . . . . 14 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ( I ↾ (Base‘𝐺)) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
10956, 108syl5eq 2667 . . . . . . . . . . . . 13 ((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → (𝑥 ∈ (Base‘𝐺) ↦ 𝑥) = (𝑥 ∈ (Base‘𝐺) ↦ ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
110 mpteqb 6297 . . . . . . . . . . . . . 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 2925 . . . . . . . . . . . . 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 2931 . . . . . . . . . . 11 (((𝐼 ≈ 1𝑜 ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
115114an32s 846 . . . . . . . . . 10 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
11670eqeq2d 2631 . . . . . . . . . . 11 (𝑛 = (𝑓𝑥) → (𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)) ↔ 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))))
117116rspcev 3307 . . . . . . . . . 10 (((𝑓𝑥) ∈ ℤ ∧ 𝑥 = ((𝑓𝑥)(.g𝐺)((varFGrp𝐼)‘ 𝐼))) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
11855, 115, 117syl2anc 693 . . . . . . . . 9 (((𝐼 ≈ 1𝑜𝑥 ∈ (Base‘𝐺)) ∧ (𝑓 ∈ (𝐺 GrpHom ℤring) ∧ (𝑓‘((varFGrp𝐼)‘ 𝐼)) = 1)) → ∃𝑛 ∈ ℤ 𝑥 = (𝑛(.g𝐺)((varFGrp𝐼)‘ 𝐼)))
119118expr 643 . . . . . . . 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 3029 . . . . . 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 18283 . . . 4 (𝐼 ≈ 1𝑜𝐺 ∈ CycGrp)
12415, 123jaoi 394 . . 3 ((𝐼 ≺ 1𝑜𝐼 ≈ 1𝑜) → 𝐺 ∈ CycGrp)
1251, 124sylbi 207 . 2 (𝐼 ≼ 1𝑜𝐺 ∈ CycGrp)
126 cygabl 18286 . . 3 (𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
1273frgpnabl 18272 . . . . 5 (1𝑜𝐼 → ¬ 𝐺 ∈ Abel)
128127con2i 134 . . . 4 (𝐺 ∈ Abel → ¬ 1𝑜𝐼)
129 ablgrp 18192 . . . . . 6 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
130 eqid 2621 . . . . . . 7 (0g𝐺) = (0g𝐺)
13116, 130grpidcl 17444 . . . . . 6 (𝐺 ∈ Grp → (0g𝐺) ∈ (Base‘𝐺))
1323, 16elbasfv 15914 . . . . . 6 ((0g𝐺) ∈ (Base‘𝐺) → 𝐼 ∈ V)
133129, 131, 1323syl 18 . . . . 5 (𝐺 ∈ Abel → 𝐼 ∈ V)
134 1onn 7716 . . . . . 6 1𝑜 ∈ ω
135 nnfi 8150 . . . . . 6 (1𝑜 ∈ ω → 1𝑜 ∈ Fin)
136134, 135ax-mp 5 . . . . 5 1𝑜 ∈ Fin
137 fidomtri2 8817 . . . . 5 ((𝐼 ∈ V ∧ 1𝑜 ∈ Fin) → (𝐼 ≼ 1𝑜 ↔ ¬ 1𝑜𝐼))
138133, 136, 137sylancl 694 . . . 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 383  wa 384   = wceq 1482  wcel 1989  wral 2911  wrex 2912  ∃!wreu 2913  ∃*wrmo 2914  Vcvv 3198  c0 3913  {csn 4175  cop 4181   cuni 4434   class class class wbr 4651  cmpt 4727   I cid 5021  cres 5114  ccom 5116  wf 5882  cfv 5886  (class class class)co 6647  ωcom 7062  1𝑜c1o 7550  cen 7949  cdom 7950  csdm 7951  Fincfn 7952  1c1 9934  cz 11374  Basecbs 15851  0gc0g 16094  Grpcgrp 17416  .gcmg 17534   GrpHom cghm 17651   ~FG cefg 18113  freeGrpcfrgp 18114  varFGrpcvrgp 18115  Abelcabl 18188  CycGrpccyg 18273  ringzring 19812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010  ax-addf 10012  ax-mulf 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-ot 4184  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-er 7739  df-ec 7741  df-qs 7745  df-map 7856  df-pm 7857  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-sup 8345  df-inf 8346  df-card 8762  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-3 11077  df-4 11078  df-5 11079  df-6 11080  df-7 11081  df-8 11082  df-9 11083  df-n0 11290  df-xnn0 11361  df-z 11375  df-dec 11491  df-uz 11685  df-rp 11830  df-fz 12324  df-fzo 12462  df-seq 12797  df-hash 13113  df-word 13294  df-lsw 13295  df-concat 13296  df-s1 13297  df-substr 13298  df-splice 13299  df-reverse 13300  df-s2 13587  df-struct 15853  df-ndx 15854  df-slot 15855  df-base 15857  df-sets 15858  df-ress 15859  df-plusg 15948  df-mulr 15949  df-starv 15950  df-sca 15951  df-vsca 15952  df-ip 15953  df-tset 15954  df-ple 15955  df-ds 15958  df-unif 15959  df-0g 16096  df-gsum 16097  df-imas 16162  df-qus 16163  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-mhm 17329  df-submnd 17330  df-frmd 17380  df-vrmd 17381  df-grp 17419  df-minusg 17420  df-mulg 17535  df-subg 17585  df-ghm 17652  df-efg 18116  df-frgp 18117  df-vrgp 18118  df-cmn 18189  df-abl 18190  df-cyg 18274  df-mgp 18484  df-ur 18496  df-ring 18543  df-cring 18544  df-subrg 18772  df-cnfld 19741  df-zring 19813
This theorem is referenced by: (None)
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