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Theorem cygznlem3 19837
Description: A cyclic group with 𝑛 elements is isomorphic to ℤ / 𝑛. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
cygzn.b 𝐵 = (Base‘𝐺)
cygzn.n 𝑁 = if(𝐵 ∈ Fin, (#‘𝐵), 0)
cygzn.y 𝑌 = (ℤ/nℤ‘𝑁)
cygzn.m · = (.g𝐺)
cygzn.l 𝐿 = (ℤRHom‘𝑌)
cygzn.e 𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
cygzn.g (𝜑𝐺 ∈ CycGrp)
cygzn.x (𝜑𝑋𝐸)
cygzn.f 𝐹 = ran (𝑚 ∈ ℤ ↦ ⟨(𝐿𝑚), (𝑚 · 𝑋)⟩)
Assertion
Ref Expression
cygznlem3 (𝜑𝐺𝑔 𝑌)
Distinct variable groups:   𝑚,𝑛,𝑥,𝐵   𝑚,𝐺,𝑛,𝑥   · ,𝑚,𝑛,𝑥   𝑚,𝑌,𝑛,𝑥   𝑚,𝐿,𝑛,𝑥   𝑥,𝑁   𝜑,𝑚   𝑛,𝐹,𝑥   𝑚,𝑋,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑛)   𝐸(𝑥,𝑚,𝑛)   𝐹(𝑚)   𝑁(𝑚,𝑛)

Proof of Theorem cygznlem3
Dummy variables 𝑎 𝑏 𝑖 𝑗 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . 4 (Base‘𝑌) = (Base‘𝑌)
2 cygzn.b . . . 4 𝐵 = (Base‘𝐺)
3 eqid 2621 . . . 4 (+g𝑌) = (+g𝑌)
4 eqid 2621 . . . 4 (+g𝐺) = (+g𝐺)
5 cygzn.n . . . . . 6 𝑁 = if(𝐵 ∈ Fin, (#‘𝐵), 0)
6 hashcl 13087 . . . . . . . 8 (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ0)
76adantl 482 . . . . . . 7 ((𝜑𝐵 ∈ Fin) → (#‘𝐵) ∈ ℕ0)
8 0nn0 11251 . . . . . . . 8 0 ∈ ℕ0
98a1i 11 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0)
107, 9ifclda 4092 . . . . . 6 (𝜑 → if(𝐵 ∈ Fin, (#‘𝐵), 0) ∈ ℕ0)
115, 10syl5eqel 2702 . . . . 5 (𝜑𝑁 ∈ ℕ0)
12 cygzn.y . . . . . 6 𝑌 = (ℤ/nℤ‘𝑁)
1312zncrng 19812 . . . . 5 (𝑁 ∈ ℕ0𝑌 ∈ CRing)
14 crngring 18479 . . . . 5 (𝑌 ∈ CRing → 𝑌 ∈ Ring)
15 ringgrp 18473 . . . . 5 (𝑌 ∈ Ring → 𝑌 ∈ Grp)
1611, 13, 14, 154syl 19 . . . 4 (𝜑𝑌 ∈ Grp)
17 cygzn.g . . . . 5 (𝜑𝐺 ∈ CycGrp)
18 cyggrp 18212 . . . . 5 (𝐺 ∈ CycGrp → 𝐺 ∈ Grp)
1917, 18syl 17 . . . 4 (𝜑𝐺 ∈ Grp)
20 cygzn.m . . . . 5 · = (.g𝐺)
21 cygzn.l . . . . 5 𝐿 = (ℤRHom‘𝑌)
22 cygzn.e . . . . 5 𝐸 = {𝑥𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}
23 cygzn.x . . . . 5 (𝜑𝑋𝐸)
24 cygzn.f . . . . 5 𝐹 = ran (𝑚 ∈ ℤ ↦ ⟨(𝐿𝑚), (𝑚 · 𝑋)⟩)
252, 5, 12, 20, 21, 22, 17, 23, 24cygznlem2a 19835 . . . 4 (𝜑𝐹:(Base‘𝑌)⟶𝐵)
2612, 1, 21znzrhfo 19815 . . . . . . . 8 (𝑁 ∈ ℕ0𝐿:ℤ–onto→(Base‘𝑌))
2711, 26syl 17 . . . . . . 7 (𝜑𝐿:ℤ–onto→(Base‘𝑌))
28 foelrn 6334 . . . . . . 7 ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖))
2927, 28sylan 488 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖))
30 foelrn 6334 . . . . . . 7 ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗))
3127, 30sylan 488 . . . . . 6 ((𝜑𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗))
3229, 31anim12dan 881 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗)))
33 reeanv 3097 . . . . . . 7 (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) ↔ (∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗)))
3419adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐺 ∈ Grp)
35 simprl 793 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑖 ∈ ℤ)
36 simprr 795 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑗 ∈ ℤ)
372, 20, 22iscyggen 18203 . . . . . . . . . . . . . 14 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
3837simplbi 476 . . . . . . . . . . . . 13 (𝑋𝐸𝑋𝐵)
3923, 38syl 17 . . . . . . . . . . . 12 (𝜑𝑋𝐵)
4039adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑋𝐵)
412, 20, 4mulgdir 17494 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g𝐺)(𝑗 · 𝑋)))
4234, 35, 36, 40, 41syl13anc 1325 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g𝐺)(𝑗 · 𝑋)))
4311, 13syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ CRing)
4421zrhrhm 19779 . . . . . . . . . . . . . . 15 (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌))
45 rhmghm 18646 . . . . . . . . . . . . . . 15 (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿 ∈ (ℤring GrpHom 𝑌))
4643, 14, 44, 454syl 19 . . . . . . . . . . . . . 14 (𝜑𝐿 ∈ (ℤring GrpHom 𝑌))
4746adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐿 ∈ (ℤring GrpHom 𝑌))
48 zringbas 19743 . . . . . . . . . . . . . 14 ℤ = (Base‘ℤring)
49 zringplusg 19744 . . . . . . . . . . . . . 14 + = (+g‘ℤring)
5048, 49, 3ghmlin 17586 . . . . . . . . . . . . 13 ((𝐿 ∈ (ℤring GrpHom 𝑌) ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿𝑖)(+g𝑌)(𝐿𝑗)))
5147, 35, 36, 50syl3anc 1323 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿𝑖)(+g𝑌)(𝐿𝑗)))
5251fveq2d 6152 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))))
53 zaddcl 11361 . . . . . . . . . . . 12 ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 + 𝑗) ∈ ℤ)
542, 5, 12, 20, 21, 22, 17, 23, 24cygznlem2 19836 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 + 𝑗) ∈ ℤ) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
5553, 54sylan2 491 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
5652, 55eqtr3d 2657 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))) = ((𝑖 + 𝑗) · 𝑋))
572, 5, 12, 20, 21, 22, 17, 23, 24cygznlem2 19836 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ℤ) → (𝐹‘(𝐿𝑖)) = (𝑖 · 𝑋))
5857adantrr 752 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿𝑖)) = (𝑖 · 𝑋))
592, 5, 12, 20, 21, 22, 17, 23, 24cygznlem2 19836 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℤ) → (𝐹‘(𝐿𝑗)) = (𝑗 · 𝑋))
6059adantrl 751 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿𝑗)) = (𝑗 · 𝑋))
6158, 60oveq12d 6622 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗))) = ((𝑖 · 𝑋)(+g𝐺)(𝑗 · 𝑋)))
6242, 56, 613eqtr4d 2665 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))) = ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗))))
63 oveq12 6613 . . . . . . . . . . 11 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝑎(+g𝑌)𝑏) = ((𝐿𝑖)(+g𝑌)(𝐿𝑗)))
6463fveq2d 6152 . . . . . . . . . 10 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝐹‘(𝑎(+g𝑌)𝑏)) = (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))))
65 fveq2 6148 . . . . . . . . . . 11 (𝑎 = (𝐿𝑖) → (𝐹𝑎) = (𝐹‘(𝐿𝑖)))
66 fveq2 6148 . . . . . . . . . . 11 (𝑏 = (𝐿𝑗) → (𝐹𝑏) = (𝐹‘(𝐿𝑗)))
6765, 66oveqan12d 6623 . . . . . . . . . 10 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎)(+g𝐺)(𝐹𝑏)) = ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗))))
6864, 67eqeq12d 2636 . . . . . . . . 9 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏)) ↔ (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))) = ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗)))))
6962, 68syl5ibrcom 237 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏))))
7069rexlimdvva 3031 . . . . . . 7 (𝜑 → (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏))))
7133, 70syl5bir 233 . . . . . 6 (𝜑 → ((∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗)) → (𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏))))
7271imp 445 . . . . 5 ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗))) → (𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏)))
7332, 72syldan 487 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏)))
741, 2, 3, 4, 16, 19, 25, 73isghmd 17590 . . 3 (𝜑𝐹 ∈ (𝑌 GrpHom 𝐺))
7558, 60eqeq12d 2636 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗)) ↔ (𝑖 · 𝑋) = (𝑗 · 𝑋)))
762, 5, 12, 20, 21, 22, 17, 23cygznlem1 19834 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐿𝑖) = (𝐿𝑗) ↔ (𝑖 · 𝑋) = (𝑗 · 𝑋)))
7775, 76bitr4d 271 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗)) ↔ (𝐿𝑖) = (𝐿𝑗)))
7877biimpd 219 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗)) → (𝐿𝑖) = (𝐿𝑗)))
7965, 66eqeqan12d 2637 . . . . . . . . . . . 12 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎) = (𝐹𝑏) ↔ (𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗))))
80 eqeq12 2634 . . . . . . . . . . . 12 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝑎 = 𝑏 ↔ (𝐿𝑖) = (𝐿𝑗)))
8179, 80imbi12d 334 . . . . . . . . . . 11 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏) ↔ ((𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗)) → (𝐿𝑖) = (𝐿𝑗))))
8278, 81syl5ibrcom 237 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
8382rexlimdvva 3031 . . . . . . . . 9 (𝜑 → (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
8433, 83syl5bir 233 . . . . . . . 8 (𝜑 → ((∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
8584imp 445 . . . . . . 7 ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗))) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
8632, 85syldan 487 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
8786ralrimivva 2965 . . . . 5 (𝜑 → ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
88 dff13 6466 . . . . 5 (𝐹:(Base‘𝑌)–1-1𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
8925, 87, 88sylanbrc 697 . . . 4 (𝜑𝐹:(Base‘𝑌)–1-1𝐵)
902, 20, 22iscyggen2 18204 . . . . . . . . 9 (𝐺 ∈ Grp → (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))))
9119, 90syl 17 . . . . . . . 8 (𝜑 → (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))))
9223, 91mpbid 222 . . . . . . 7 (𝜑 → (𝑋𝐵 ∧ ∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋)))
9392simprd 479 . . . . . 6 (𝜑 → ∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))
94 oveq1 6611 . . . . . . . . . 10 (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
9594eqeq2d 2631 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑧 = (𝑛 · 𝑋) ↔ 𝑧 = (𝑗 · 𝑋)))
9695cbvrexv 3160 . . . . . . . 8 (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) ↔ ∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋))
9727adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝐿:ℤ–onto→(Base‘𝑌))
98 fof 6072 . . . . . . . . . . . . 13 (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
9997, 98syl 17 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → 𝐿:ℤ⟶(Base‘𝑌))
10099ffvelrnda 6315 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝐿𝑗) ∈ (Base‘𝑌))
10159adantlr 750 . . . . . . . . . . . 12 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿𝑗)) = (𝑗 · 𝑋))
102101eqcomd 2627 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝑗 · 𝑋) = (𝐹‘(𝐿𝑗)))
103 fveq2 6148 . . . . . . . . . . . . 13 (𝑎 = (𝐿𝑗) → (𝐹𝑎) = (𝐹‘(𝐿𝑗)))
104103eqeq2d 2631 . . . . . . . . . . . 12 (𝑎 = (𝐿𝑗) → ((𝑗 · 𝑋) = (𝐹𝑎) ↔ (𝑗 · 𝑋) = (𝐹‘(𝐿𝑗))))
105104rspcev 3295 . . . . . . . . . . 11 (((𝐿𝑗) ∈ (Base‘𝑌) ∧ (𝑗 · 𝑋) = (𝐹‘(𝐿𝑗))) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹𝑎))
106100, 102, 105syl2anc 692 . . . . . . . . . 10 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹𝑎))
107 eqeq1 2625 . . . . . . . . . . 11 (𝑧 = (𝑗 · 𝑋) → (𝑧 = (𝐹𝑎) ↔ (𝑗 · 𝑋) = (𝐹𝑎)))
108107rexbidv 3045 . . . . . . . . . 10 (𝑧 = (𝑗 · 𝑋) → (∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎) ↔ ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹𝑎)))
109106, 108syl5ibrcom 237 . . . . . . . . 9 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝑧 = (𝑗 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
110109rexlimdva 3024 . . . . . . . 8 ((𝜑𝑧𝐵) → (∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
11196, 110syl5bi 232 . . . . . . 7 ((𝜑𝑧𝐵) → (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
112111ralimdva 2956 . . . . . 6 (𝜑 → (∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∀𝑧𝐵𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
11393, 112mpd 15 . . . . 5 (𝜑 → ∀𝑧𝐵𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎))
114 dffo3 6330 . . . . 5 (𝐹:(Base‘𝑌)–onto𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑧𝐵𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
11525, 113, 114sylanbrc 697 . . . 4 (𝜑𝐹:(Base‘𝑌)–onto𝐵)
116 df-f1o 5854 . . . 4 (𝐹:(Base‘𝑌)–1-1-onto𝐵 ↔ (𝐹:(Base‘𝑌)–1-1𝐵𝐹:(Base‘𝑌)–onto𝐵))
11789, 115, 116sylanbrc 697 . . 3 (𝜑𝐹:(Base‘𝑌)–1-1-onto𝐵)
1181, 2isgim 17625 . . 3 (𝐹 ∈ (𝑌 GrpIso 𝐺) ↔ (𝐹 ∈ (𝑌 GrpHom 𝐺) ∧ 𝐹:(Base‘𝑌)–1-1-onto𝐵))
11974, 117, 118sylanbrc 697 . 2 (𝜑𝐹 ∈ (𝑌 GrpIso 𝐺))
120 brgici 17633 . 2 (𝐹 ∈ (𝑌 GrpIso 𝐺) → 𝑌𝑔 𝐺)
121 gicsym 17637 . 2 (𝑌𝑔 𝐺𝐺𝑔 𝑌)
122119, 120, 1213syl 18 1 (𝜑𝐺𝑔 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  ifcif 4058  cop 4154   class class class wbr 4613  cmpt 4673  ran crn 5075  wf 5843  1-1wf1 5844  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  Fincfn 7899  0cc0 9880   + caddc 9883  0cn0 11236  cz 11321  #chash 13057  Basecbs 15781  +gcplusg 15862  Grpcgrp 17343  .gcmg 17461   GrpHom cghm 17578   GrpIso cgim 17620  𝑔 cgic 17621  CycGrpccyg 18200  Ringcrg 18468  CRingccrg 18469   RingHom crh 18633  ringzring 19737  ℤRHomczrh 19767  ℤ/nczn 19770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-tpos 7297  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-omul 7510  df-er 7687  df-ec 7689  df-qs 7693  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-acn 8712  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-rp 11777  df-fz 12269  df-fl 12533  df-mod 12609  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-dvds 14908  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-0g 16023  df-imas 16089  df-qus 16090  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mulg 17462  df-subg 17512  df-nsg 17513  df-eqg 17514  df-ghm 17579  df-gim 17622  df-gic 17623  df-od 17869  df-cmn 18116  df-abl 18117  df-cyg 18201  df-mgp 18411  df-ur 18423  df-ring 18470  df-cring 18471  df-oppr 18544  df-dvdsr 18562  df-rnghom 18636  df-subrg 18699  df-lmod 18786  df-lss 18852  df-lsp 18891  df-sra 19091  df-rgmod 19092  df-lidl 19093  df-rsp 19094  df-2idl 19151  df-cnfld 19666  df-zring 19738  df-zrh 19771  df-zn 19774
This theorem is referenced by:  cygzn  19838
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