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Theorem cygznlem3 20120
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 2760 . . . 4 (Base‘𝑌) = (Base‘𝑌)
2 cygzn.b . . . 4 𝐵 = (Base‘𝐺)
3 eqid 2760 . . . 4 (+g𝑌) = (+g𝑌)
4 eqid 2760 . . . 4 (+g𝐺) = (+g𝐺)
5 cygzn.n . . . . . 6 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)
6 hashcl 13339 . . . . . . . 8 (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0)
76adantl 473 . . . . . . 7 ((𝜑𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0)
8 0nn0 11499 . . . . . . . 8 0 ∈ ℕ0
98a1i 11 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0)
107, 9ifclda 4264 . . . . . 6 (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0)
115, 10syl5eqel 2843 . . . . 5 (𝜑𝑁 ∈ ℕ0)
12 cygzn.y . . . . . 6 𝑌 = (ℤ/nℤ‘𝑁)
1312zncrng 20095 . . . . 5 (𝑁 ∈ ℕ0𝑌 ∈ CRing)
14 crngring 18758 . . . . 5 (𝑌 ∈ CRing → 𝑌 ∈ Ring)
15 ringgrp 18752 . . . . 5 (𝑌 ∈ Ring → 𝑌 ∈ Grp)
1611, 13, 14, 154syl 19 . . . 4 (𝜑𝑌 ∈ Grp)
17 cygzn.g . . . . 5 (𝜑𝐺 ∈ CycGrp)
18 cyggrp 18491 . . . . 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 20118 . . . 4 (𝜑𝐹:(Base‘𝑌)⟶𝐵)
2612, 1, 21znzrhfo 20098 . . . . . . . 8 (𝑁 ∈ ℕ0𝐿:ℤ–onto→(Base‘𝑌))
2711, 26syl 17 . . . . . . 7 (𝜑𝐿:ℤ–onto→(Base‘𝑌))
28 foelrn 6541 . . . . . . 7 ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖))
2927, 28sylan 489 . . . . . 6 ((𝜑𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖))
30 foelrn 6541 . . . . . . 7 ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗))
3127, 30sylan 489 . . . . . 6 ((𝜑𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗))
3229, 31anim12dan 918 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗)))
33 reeanv 3245 . . . . . . 7 (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) ↔ (∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗)))
3419adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐺 ∈ Grp)
35 simprl 811 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑖 ∈ ℤ)
36 simprr 813 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑗 ∈ ℤ)
372, 20, 22iscyggen 18482 . . . . . . . . . . . . . 14 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵))
3837simplbi 478 . . . . . . . . . . . . 13 (𝑋𝐸𝑋𝐵)
3923, 38syl 17 . . . . . . . . . . . 12 (𝜑𝑋𝐵)
4039adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑋𝐵)
412, 20, 4mulgdir 17774 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g𝐺)(𝑗 · 𝑋)))
4234, 35, 36, 40, 41syl13anc 1479 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g𝐺)(𝑗 · 𝑋)))
4311, 13syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ CRing)
4421zrhrhm 20062 . . . . . . . . . . . . . . 15 (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌))
45 rhmghm 18927 . . . . . . . . . . . . . . 15 (𝐿 ∈ (ℤring RingHom 𝑌) → 𝐿 ∈ (ℤring GrpHom 𝑌))
4643, 14, 44, 454syl 19 . . . . . . . . . . . . . 14 (𝜑𝐿 ∈ (ℤring GrpHom 𝑌))
4746adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐿 ∈ (ℤring GrpHom 𝑌))
48 zringbas 20026 . . . . . . . . . . . . . 14 ℤ = (Base‘ℤring)
49 zringplusg 20027 . . . . . . . . . . . . . 14 + = (+g‘ℤring)
5048, 49, 3ghmlin 17866 . . . . . . . . . . . . 13 ((𝐿 ∈ (ℤring GrpHom 𝑌) ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿𝑖)(+g𝑌)(𝐿𝑗)))
5147, 35, 36, 50syl3anc 1477 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿𝑖)(+g𝑌)(𝐿𝑗)))
5251fveq2d 6356 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))))
53 zaddcl 11609 . . . . . . . . . . . 12 ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 + 𝑗) ∈ ℤ)
542, 5, 12, 20, 21, 22, 17, 23, 24cygznlem2 20119 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 + 𝑗) ∈ ℤ) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
5553, 54sylan2 492 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋))
5652, 55eqtr3d 2796 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))) = ((𝑖 + 𝑗) · 𝑋))
572, 5, 12, 20, 21, 22, 17, 23, 24cygznlem2 20119 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ ℤ) → (𝐹‘(𝐿𝑖)) = (𝑖 · 𝑋))
5857adantrr 755 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿𝑖)) = (𝑖 · 𝑋))
592, 5, 12, 20, 21, 22, 17, 23, 24cygznlem2 20119 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℤ) → (𝐹‘(𝐿𝑗)) = (𝑗 · 𝑋))
6059adantrl 754 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿𝑗)) = (𝑗 · 𝑋))
6158, 60oveq12d 6831 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗))) = ((𝑖 · 𝑋)(+g𝐺)(𝑗 · 𝑋)))
6242, 56, 613eqtr4d 2804 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))) = ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗))))
63 oveq12 6822 . . . . . . . . . . 11 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝑎(+g𝑌)𝑏) = ((𝐿𝑖)(+g𝑌)(𝐿𝑗)))
6463fveq2d 6356 . . . . . . . . . 10 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝐹‘(𝑎(+g𝑌)𝑏)) = (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))))
65 fveq2 6352 . . . . . . . . . . 11 (𝑎 = (𝐿𝑖) → (𝐹𝑎) = (𝐹‘(𝐿𝑖)))
66 fveq2 6352 . . . . . . . . . . 11 (𝑏 = (𝐿𝑗) → (𝐹𝑏) = (𝐹‘(𝐿𝑗)))
6765, 66oveqan12d 6832 . . . . . . . . . 10 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎)(+g𝐺)(𝐹𝑏)) = ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗))))
6864, 67eqeq12d 2775 . . . . . . . . 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 3176 . . . . . . 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 444 . . . . 5 ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗))) → (𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏)))
7332, 72syldan 488 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝐹‘(𝑎(+g𝑌)𝑏)) = ((𝐹𝑎)(+g𝐺)(𝐹𝑏)))
741, 2, 3, 4, 16, 19, 25, 73isghmd 17870 . . 3 (𝜑𝐹 ∈ (𝑌 GrpHom 𝐺))
7558, 60eqeq12d 2775 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗)) ↔ (𝑖 · 𝑋) = (𝑗 · 𝑋)))
762, 5, 12, 20, 21, 22, 17, 23cygznlem1 20117 . . . . . . . . . . . . 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 2776 . . . . . . . . . . . 12 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎) = (𝐹𝑏) ↔ (𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗))))
80 eqeq12 2773 . . . . . . . . . . . 12 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝑎 = 𝑏 ↔ (𝐿𝑖) = (𝐿𝑗)))
8179, 80imbi12d 333 . . . . . . . . . . 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 3176 . . . . . . . . 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 444 . . . . . . 7 ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿𝑗))) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
8632, 85syldan 488 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
8786ralrimivva 3109 . . . . 5 (𝜑 → ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
88 dff13 6675 . . . . 5 (𝐹:(Base‘𝑌)–1-1𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
8925, 87, 88sylanbrc 701 . . . 4 (𝜑𝐹:(Base‘𝑌)–1-1𝐵)
902, 20, 22iscyggen2 18483 . . . . . . . . 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 482 . . . . . 6 (𝜑 → ∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))
94 oveq1 6820 . . . . . . . . . 10 (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
9594eqeq2d 2770 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑧 = (𝑛 · 𝑋) ↔ 𝑧 = (𝑗 · 𝑋)))
9695cbvrexv 3311 . . . . . . . 8 (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) ↔ ∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋))
9727adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝐿:ℤ–onto→(Base‘𝑌))
98 fof 6276 . . . . . . . . . . . . 13 (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
9997, 98syl 17 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → 𝐿:ℤ⟶(Base‘𝑌))
10099ffvelrnda 6522 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝐿𝑗) ∈ (Base‘𝑌))
10159adantlr 753 . . . . . . . . . . . 12 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿𝑗)) = (𝑗 · 𝑋))
102101eqcomd 2766 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝑗 · 𝑋) = (𝐹‘(𝐿𝑗)))
103 fveq2 6352 . . . . . . . . . . . . 13 (𝑎 = (𝐿𝑗) → (𝐹𝑎) = (𝐹‘(𝐿𝑗)))
104103eqeq2d 2770 . . . . . . . . . . . 12 (𝑎 = (𝐿𝑗) → ((𝑗 · 𝑋) = (𝐹𝑎) ↔ (𝑗 · 𝑋) = (𝐹‘(𝐿𝑗))))
105104rspcev 3449 . . . . . . . . . . 11 (((𝐿𝑗) ∈ (Base‘𝑌) ∧ (𝑗 · 𝑋) = (𝐹‘(𝐿𝑗))) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹𝑎))
106100, 102, 105syl2anc 696 . . . . . . . . . 10 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹𝑎))
107 eqeq1 2764 . . . . . . . . . . 11 (𝑧 = (𝑗 · 𝑋) → (𝑧 = (𝐹𝑎) ↔ (𝑗 · 𝑋) = (𝐹𝑎)))
108107rexbidv 3190 . . . . . . . . . 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 3169 . . . . . . . 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 3100 . . . . . 6 (𝜑 → (∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∀𝑧𝐵𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
11393, 112mpd 15 . . . . 5 (𝜑 → ∀𝑧𝐵𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎))
114 dffo3 6537 . . . . 5 (𝐹:(Base‘𝑌)–onto𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑧𝐵𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
11525, 113, 114sylanbrc 701 . . . 4 (𝜑𝐹:(Base‘𝑌)–onto𝐵)
116 df-f1o 6056 . . . 4 (𝐹:(Base‘𝑌)–1-1-onto𝐵 ↔ (𝐹:(Base‘𝑌)–1-1𝐵𝐹:(Base‘𝑌)–onto𝐵))
11789, 115, 116sylanbrc 701 . . 3 (𝜑𝐹:(Base‘𝑌)–1-1-onto𝐵)
1181, 2isgim 17905 . . 3 (𝐹 ∈ (𝑌 GrpIso 𝐺) ↔ (𝐹 ∈ (𝑌 GrpHom 𝐺) ∧ 𝐹:(Base‘𝑌)–1-1-onto𝐵))
11974, 117, 118sylanbrc 701 . 2 (𝜑𝐹 ∈ (𝑌 GrpIso 𝐺))
120 brgici 17913 . 2 (𝐹 ∈ (𝑌 GrpIso 𝐺) → 𝑌𝑔 𝐺)
121 gicsym 17917 . 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 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  ifcif 4230  cop 4327   class class class wbr 4804  cmpt 4881  ran crn 5267  wf 6045  1-1wf1 6046  ontowfo 6047  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  Fincfn 8121  0cc0 10128   + caddc 10131  0cn0 11484  cz 11569  chash 13311  Basecbs 16059  +gcplusg 16143  Grpcgrp 17623  .gcmg 17741   GrpHom cghm 17858   GrpIso cgim 17900  𝑔 cgic 17901  CycGrpccyg 18479  Ringcrg 18747  CRingccrg 18748   RingHom crh 18914  ringzring 20020  ℤRHomczrh 20050  ℤ/nczn 20053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-addf 10207  ax-mulf 10208
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-tpos 7521  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-omul 7734  df-er 7911  df-ec 7913  df-qs 7917  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-acn 8958  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-rp 12026  df-fz 12520  df-fl 12787  df-mod 12863  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-dvds 15183  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-starv 16158  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-unif 16167  df-0g 16304  df-imas 16370  df-qus 16371  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-grp 17626  df-minusg 17627  df-sbg 17628  df-mulg 17742  df-subg 17792  df-nsg 17793  df-eqg 17794  df-ghm 17859  df-gim 17902  df-gic 17903  df-od 18148  df-cmn 18395  df-abl 18396  df-cyg 18480  df-mgp 18690  df-ur 18702  df-ring 18749  df-cring 18750  df-oppr 18823  df-dvdsr 18841  df-rnghom 18917  df-subrg 18980  df-lmod 19067  df-lss 19135  df-lsp 19174  df-sra 19374  df-rgmod 19375  df-lidl 19376  df-rsp 19377  df-2idl 19434  df-cnfld 19949  df-zring 20021  df-zrh 20054  df-zn 20057
This theorem is referenced by:  cygzn  20121
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