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Theorem cygznlem3 21589
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 2736 . . . 4 (Base‘𝑌) = (Base‘𝑌)
2 cygzn.b . . . 4 𝐵 = (Base‘𝐺)
3 eqid 2736 . . . 4 (+g𝑌) = (+g𝑌)
4 eqid 2736 . . . 4 (+g𝐺) = (+g𝐺)
5 cygzn.n . . . . . 6 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)
6 hashcl 14396 . . . . . . . 8 (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0)
76adantl 481 . . . . . . 7 ((𝜑𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0)
8 0nn0 12543 . . . . . . . 8 0 ∈ ℕ0
98a1i 11 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0)
107, 9ifclda 4560 . . . . . 6 (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0)
115, 10eqeltrid 2844 . . . . 5 (𝜑𝑁 ∈ ℕ0)
12 cygzn.y . . . . . 6 𝑌 = (ℤ/nℤ‘𝑁)
1312zncrng 21564 . . . . 5 (𝑁 ∈ ℕ0𝑌 ∈ CRing)
14 crngring 20243 . . . . 5 (𝑌 ∈ CRing → 𝑌 ∈ Ring)
15 ringgrp 20236 . . . . 5 (𝑌 ∈ Ring → 𝑌 ∈ Grp)
1611, 13, 14, 154syl 19 . . . 4 (𝜑𝑌 ∈ Grp)
17 cygzn.g . . . . 5 (𝜑𝐺 ∈ CycGrp)
18 cyggrp 19909 . . . . 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 21587 . . . 4 (𝜑𝐹:(Base‘𝑌)⟶𝐵)
2612, 1, 21znzrhfo 21567 . . . . . . . 8 (𝑁 ∈ ℕ0𝐿:ℤ–onto→(Base‘𝑌))
2711, 26syl 17 . . . . . . 7 (𝜑𝐿:ℤ–onto→(Base‘𝑌))
28 foelrn 7126 . . . . . . 7 ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖))
2927, 28sylan 580 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖))
30 foelrn 7126 . . . . . . 7 ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗))
3127, 30sylan 580 . . . . . 6 ((𝜑𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗))
3229, 31anim12dan 619 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗)))
33 reeanv 3228 . . . . . . 7 (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) ↔ (∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗)))
3419adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐺 ∈ Grp)
35 simprl 770 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑖 ∈ ℤ)
36 simprr 772 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑗 ∈ ℤ)
372, 20, 22iscyggen 19899 . . . . . . . . . . . . . 14 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
3837simplbi 497 . . . . . . . . . . . . 13 (𝑋𝐸𝑋𝐵)
3923, 38syl 17 . . . . . . . . . . . 12 (𝜑𝑋𝐵)
4039adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑋𝐵)
412, 20, 4mulgdir 19125 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g𝐺)(𝑗 · 𝑋)))
4234, 35, 36, 40, 41syl13anc 1373 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g𝐺)(𝑗 · 𝑋)))
4311, 13syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ CRing)
4421zrhrhm 21523 . . . . . . . . . . . . . . 15 (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌))
45 rhmghm 20485 . . . . . . . . . . . . . . 15 (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿 ∈ (ℤring GrpHom 𝑌))
4643, 14, 44, 454syl 19 . . . . . . . . . . . . . 14 (𝜑𝐿 ∈ (ℤring GrpHom 𝑌))
4746adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐿 ∈ (ℤring GrpHom 𝑌))
48 zringbas 21465 . . . . . . . . . . . . . 14 ℤ = (Base‘ℤring)
49 zringplusg 21466 . . . . . . . . . . . . . 14 + = (+g‘ℤring)
5048, 49, 3ghmlin 19240 . . . . . . . . . . . . 13 ((𝐿 ∈ (ℤring GrpHom 𝑌) ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿𝑖)(+g𝑌)(𝐿𝑗)))
5147, 35, 36, 50syl3anc 1372 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿𝑖)(+g𝑌)(𝐿𝑗)))
5251fveq2d 6909 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))))
53 zaddcl 12659 . . . . . . . . . . . 12 ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 + 𝑗) ∈ ℤ)
542, 5, 12, 20, 21, 22, 17, 23, 24cygznlem2 21588 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 + 𝑗) ∈ ℤ) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
5553, 54sylan2 593 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
5652, 55eqtr3d 2778 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))) = ((𝑖 + 𝑗) · 𝑋))
572, 5, 12, 20, 21, 22, 17, 23, 24cygznlem2 21588 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ℤ) → (𝐹‘(𝐿𝑖)) = (𝑖 · 𝑋))
5857adantrr 717 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿𝑖)) = (𝑖 · 𝑋))
592, 5, 12, 20, 21, 22, 17, 23, 24cygznlem2 21588 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℤ) → (𝐹‘(𝐿𝑗)) = (𝑗 · 𝑋))
6059adantrl 716 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿𝑗)) = (𝑗 · 𝑋))
6158, 60oveq12d 7450 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗))) = ((𝑖 · 𝑋)(+g𝐺)(𝑗 · 𝑋)))
6242, 56, 613eqtr4d 2786 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))) = ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗))))
63 oveq12 7441 . . . . . . . . . . 11 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝑎(+g𝑌)𝑏) = ((𝐿𝑖)(+g𝑌)(𝐿𝑗)))
6463fveq2d 6909 . . . . . . . . . 10 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝐹‘(𝑎(+g𝑌)𝑏)) = (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))))
65 fveq2 6905 . . . . . . . . . . 11 (𝑎 = (𝐿𝑖) → (𝐹𝑎) = (𝐹‘(𝐿𝑖)))
66 fveq2 6905 . . . . . . . . . . 11 (𝑏 = (𝐿𝑗) → (𝐹𝑏) = (𝐹‘(𝐿𝑗)))
6765, 66oveqan12d 7451 . . . . . . . . . 10 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎)(+g𝐺)(𝐹𝑏)) = ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗))))
6864, 67eqeq12d 2752 . . . . . . . . 9 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏)) ↔ (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))) = ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗)))))
6962, 68syl5ibrcom 247 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏))))
7069rexlimdvva 3212 . . . . . . 7 (𝜑 → (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏))))
7133, 70biimtrrid 243 . . . . . 6 (𝜑 → ((∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗)) → (𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏))))
7271imp 406 . . . . 5 ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗))) → (𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏)))
7332, 72syldan 591 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏)))
741, 2, 3, 4, 16, 19, 25, 73isghmd 19244 . . 3 (𝜑𝐹 ∈ (𝑌 GrpHom 𝐺))
7558, 60eqeq12d 2752 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗)) ↔ (𝑖 · 𝑋) = (𝑗 · 𝑋)))
762, 5, 12, 20, 21, 22, 17, 23cygznlem1 21586 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐿𝑖) = (𝐿𝑗) ↔ (𝑖 · 𝑋) = (𝑗 · 𝑋)))
7775, 76bitr4d 282 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗)) ↔ (𝐿𝑖) = (𝐿𝑗)))
7877biimpd 229 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗)) → (𝐿𝑖) = (𝐿𝑗)))
7965, 66eqeqan12d 2750 . . . . . . . . . . . 12 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎) = (𝐹𝑏) ↔ (𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗))))
80 eqeq12 2753 . . . . . . . . . . . 12 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝑎 = 𝑏 ↔ (𝐿𝑖) = (𝐿𝑗)))
8179, 80imbi12d 344 . . . . . . . . . . 11 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏) ↔ ((𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗)) → (𝐿𝑖) = (𝐿𝑗))))
8278, 81syl5ibrcom 247 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
8382rexlimdvva 3212 . . . . . . . . 9 (𝜑 → (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
8433, 83biimtrrid 243 . . . . . . . 8 (𝜑 → ((∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
8584imp 406 . . . . . . 7 ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗))) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
8632, 85syldan 591 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
8786ralrimivva 3201 . . . . 5 (𝜑 → ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
88 dff13 7276 . . . . 5 (𝐹:(Base‘𝑌)–1-1𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
8925, 87, 88sylanbrc 583 . . . 4 (𝜑𝐹:(Base‘𝑌)–1-1𝐵)
902, 20, 22iscyggen2 19900 . . . . . . . . 9 (𝐺 ∈ Grp → (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))))
9119, 90syl 17 . . . . . . . 8 (𝜑 → (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))))
9223, 91mpbid 232 . . . . . . 7 (𝜑 → (𝑋𝐵 ∧ ∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋)))
9392simprd 495 . . . . . 6 (𝜑 → ∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))
94 oveq1 7439 . . . . . . . . . 10 (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
9594eqeq2d 2747 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑧 = (𝑛 · 𝑋) ↔ 𝑧 = (𝑗 · 𝑋)))
9695cbvrexvw 3237 . . . . . . . 8 (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) ↔ ∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋))
9727adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝐿:ℤ–onto→(Base‘𝑌))
98 fof 6819 . . . . . . . . . . . . 13 (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
9997, 98syl 17 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → 𝐿:ℤ⟶(Base‘𝑌))
10099ffvelcdmda 7103 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝐿𝑗) ∈ (Base‘𝑌))
10159adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿𝑗)) = (𝑗 · 𝑋))
102101eqcomd 2742 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝑗 · 𝑋) = (𝐹‘(𝐿𝑗)))
103 fveq2 6905 . . . . . . . . . . . 12 (𝑎 = (𝐿𝑗) → (𝐹𝑎) = (𝐹‘(𝐿𝑗)))
104103rspceeqv 3644 . . . . . . . . . . 11 (((𝐿𝑗) ∈ (Base‘𝑌) ∧ (𝑗 · 𝑋) = (𝐹‘(𝐿𝑗))) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹𝑎))
105100, 102, 104syl2anc 584 . . . . . . . . . 10 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹𝑎))
106 eqeq1 2740 . . . . . . . . . . 11 (𝑧 = (𝑗 · 𝑋) → (𝑧 = (𝐹𝑎) ↔ (𝑗 · 𝑋) = (𝐹𝑎)))
107106rexbidv 3178 . . . . . . . . . 10 (𝑧 = (𝑗 · 𝑋) → (∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎) ↔ ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹𝑎)))
108105, 107syl5ibrcom 247 . . . . . . . . 9 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝑧 = (𝑗 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
109108rexlimdva 3154 . . . . . . . 8 ((𝜑𝑧𝐵) → (∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
11096, 109biimtrid 242 . . . . . . 7 ((𝜑𝑧𝐵) → (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
111110ralimdva 3166 . . . . . 6 (𝜑 → (∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∀𝑧𝐵𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
11293, 111mpd 15 . . . . 5 (𝜑 → ∀𝑧𝐵𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎))
113 dffo3 7121 . . . . 5 (𝐹:(Base‘𝑌)–onto𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑧𝐵𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
11425, 112, 113sylanbrc 583 . . . 4 (𝜑𝐹:(Base‘𝑌)–onto𝐵)
115 df-f1o 6567 . . . 4 (𝐹:(Base‘𝑌)–1-1-onto𝐵 ↔ (𝐹:(Base‘𝑌)–1-1𝐵𝐹:(Base‘𝑌)–onto𝐵))
11689, 114, 115sylanbrc 583 . . 3 (𝜑𝐹:(Base‘𝑌)–1-1-onto𝐵)
1171, 2isgim 19281 . . 3 (𝐹 ∈ (𝑌 GrpIso 𝐺) ↔ (𝐹 ∈ (𝑌 GrpHom 𝐺) ∧ 𝐹:(Base‘𝑌)–1-1-onto𝐵))
11874, 116, 117sylanbrc 583 . 2 (𝜑𝐹 ∈ (𝑌 GrpIso 𝐺))
119 brgici 19290 . 2 (𝐹 ∈ (𝑌 GrpIso 𝐺) → 𝑌𝑔 𝐺)
120 gicsym 19294 . 2 (𝑌𝑔 𝐺𝐺𝑔 𝑌)
121118, 119, 1203syl 18 1 (𝜑𝐺𝑔 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  wrex 3069  {crab 3435  ifcif 4524  cop 4631   class class class wbr 5142  cmpt 5224  ran crn 5685  wf 6556  1-1wf1 6557  ontowfo 6558  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  Fincfn 8986  0cc0 11156   + caddc 11159  0cn0 12528  cz 12615  chash 14370  Basecbs 17248  +gcplusg 17298  Grpcgrp 18952  .gcmg 19086   GrpHom cghm 19231   GrpIso cgim 19276  𝑔 cgic 19277  CycGrpccyg 19896  Ringcrg 20231  CRingccrg 20232   RingHom crh 20470  ringczring 21458  ℤRHomczrh 21511  ℤ/nczn 21514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234  ax-addf 11235  ax-mulf 11236
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-omul 8512  df-er 8746  df-ec 8748  df-qs 8752  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-oi 9551  df-card 9980  df-acn 9983  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-rp 13036  df-fz 13549  df-fl 13833  df-mod 13911  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-dvds 16292  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-0g 17487  df-imas 17554  df-qus 17555  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-nsg 19143  df-eqg 19144  df-ghm 19232  df-gim 19278  df-gic 19279  df-od 19547  df-cmn 19801  df-abl 19802  df-cyg 19897  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-cring 20234  df-oppr 20335  df-dvdsr 20358  df-rhm 20473  df-subrng 20547  df-subrg 20571  df-lmod 20861  df-lss 20931  df-lsp 20971  df-sra 21173  df-rgmod 21174  df-lidl 21219  df-rsp 21220  df-2idl 21261  df-cnfld 21366  df-zring 21459  df-zrh 21515  df-zn 21518
This theorem is referenced by:  cygzn  21590
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