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Theorem cygznlem3 21606
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 2735 . . . 4 (Base‘𝑌) = (Base‘𝑌)
2 cygzn.b . . . 4 𝐵 = (Base‘𝐺)
3 eqid 2735 . . . 4 (+g𝑌) = (+g𝑌)
4 eqid 2735 . . . 4 (+g𝐺) = (+g𝐺)
5 cygzn.n . . . . . 6 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)
6 hashcl 14392 . . . . . . . 8 (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0)
76adantl 481 . . . . . . 7 ((𝜑𝐵 ∈ Fin) → (♯‘𝐵) ∈ ℕ0)
8 0nn0 12539 . . . . . . . 8 0 ∈ ℕ0
98a1i 11 . . . . . . 7 ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈ ℕ0)
107, 9ifclda 4566 . . . . . 6 (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈ ℕ0)
115, 10eqeltrid 2843 . . . . 5 (𝜑𝑁 ∈ ℕ0)
12 cygzn.y . . . . . 6 𝑌 = (ℤ/nℤ‘𝑁)
1312zncrng 21581 . . . . 5 (𝑁 ∈ ℕ0𝑌 ∈ CRing)
14 crngring 20263 . . . . 5 (𝑌 ∈ CRing → 𝑌 ∈ Ring)
15 ringgrp 20256 . . . . 5 (𝑌 ∈ Ring → 𝑌 ∈ Grp)
1611, 13, 14, 154syl 19 . . . 4 (𝜑𝑌 ∈ Grp)
17 cygzn.g . . . . 5 (𝜑𝐺 ∈ CycGrp)
18 cyggrp 19923 . . . . 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 21604 . . . 4 (𝜑𝐹:(Base‘𝑌)⟶𝐵)
2612, 1, 21znzrhfo 21584 . . . . . . . 8 (𝑁 ∈ ℕ0𝐿:ℤ–onto→(Base‘𝑌))
2711, 26syl 17 . . . . . . 7 (𝜑𝐿:ℤ–onto→(Base‘𝑌))
28 foelrn 7127 . . . . . . 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 7127 . . . . . . 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 3227 . . . . . . 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 771 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑖 ∈ ℤ)
36 simprr 773 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑗 ∈ ℤ)
372, 20, 22iscyggen 19913 . . . . . . . . . . . . . 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 19137 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑋𝐵)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g𝐺)(𝑗 · 𝑋)))
4234, 35, 36, 40, 41syl13anc 1371 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g𝐺)(𝑗 · 𝑋)))
4311, 13syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ CRing)
4421zrhrhm 21540 . . . . . . . . . . . . . . 15 (𝑌 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑌))
45 rhmghm 20501 . . . . . . . . . . . . . . 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 21482 . . . . . . . . . . . . . 14 ℤ = (Base‘ℤring)
49 zringplusg 21483 . . . . . . . . . . . . . 14 + = (+g‘ℤring)
5048, 49, 3ghmlin 19252 . . . . . . . . . . . . 13 ((𝐿 ∈ (ℤring GrpHom 𝑌) ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿𝑖)(+g𝑌)(𝐿𝑗)))
5147, 35, 36, 50syl3anc 1370 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿𝑖)(+g𝑌)(𝐿𝑗)))
5251fveq2d 6911 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))))
53 zaddcl 12655 . . . . . . . . . . . 12 ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 + 𝑗) ∈ ℤ)
542, 5, 12, 20, 21, 22, 17, 23, 24cygznlem2 21605 . . . . . . . . . . . 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 2777 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))) = ((𝑖 + 𝑗) · 𝑋))
572, 5, 12, 20, 21, 22, 17, 23, 24cygznlem2 21605 . . . . . . . . . . . 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 21605 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℤ) → (𝐹‘(𝐿𝑗)) = (𝑗 · 𝑋))
6059adantrl 716 . . . . . . . . . . 11 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿𝑗)) = (𝑗 · 𝑋))
6158, 60oveq12d 7449 . . . . . . . . . 10 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗))) = ((𝑖 · 𝑋)(+g𝐺)(𝑗 · 𝑋)))
6242, 56, 613eqtr4d 2785 . . . . . . . . 9 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))) = ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗))))
63 oveq12 7440 . . . . . . . . . . 11 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝑎(+g𝑌)𝑏) = ((𝐿𝑖)(+g𝑌)(𝐿𝑗)))
6463fveq2d 6911 . . . . . . . . . 10 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → (𝐹‘(𝑎(+g𝑌)𝑏)) = (𝐹‘((𝐿𝑖)(+g𝑌)(𝐿𝑗))))
65 fveq2 6907 . . . . . . . . . . 11 (𝑎 = (𝐿𝑖) → (𝐹𝑎) = (𝐹‘(𝐿𝑖)))
66 fveq2 6907 . . . . . . . . . . 11 (𝑏 = (𝐿𝑗) → (𝐹𝑏) = (𝐹‘(𝐿𝑗)))
6765, 66oveqan12d 7450 . . . . . . . . . 10 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎)(+g𝐺)(𝐹𝑏)) = ((𝐹‘(𝐿𝑖))(+g𝐺)(𝐹‘(𝐿𝑗))))
6864, 67eqeq12d 2751 . . . . . . . . 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 3211 . . . . . . 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 19256 . . 3 (𝜑𝐹 ∈ (𝑌 GrpHom 𝐺))
7558, 60eqeq12d 2751 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗)) ↔ (𝑖 · 𝑋) = (𝑗 · 𝑋)))
762, 5, 12, 20, 21, 22, 17, 23cygznlem1 21603 . . . . . . . . . . . . 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 2749 . . . . . . . . . . . 12 ((𝑎 = (𝐿𝑖) ∧ 𝑏 = (𝐿𝑗)) → ((𝐹𝑎) = (𝐹𝑏) ↔ (𝐹‘(𝐿𝑖)) = (𝐹‘(𝐿𝑗))))
80 eqeq12 2752 . . . . . . . . . . . 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 3211 . . . . . . . . 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 3200 . . . . 5 (𝜑 → ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
88 dff13 7275 . . . . 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 19914 . . . . . . . . 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 7438 . . . . . . . . . 10 (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋))
9594eqeq2d 2746 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑧 = (𝑛 · 𝑋) ↔ 𝑧 = (𝑗 · 𝑋)))
9695cbvrexvw 3236 . . . . . . . 8 (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) ↔ ∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋))
9727adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝐿:ℤ–onto→(Base‘𝑌))
98 fof 6821 . . . . . . . . . . . . 13 (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌))
9997, 98syl 17 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → 𝐿:ℤ⟶(Base‘𝑌))
10099ffvelcdmda 7104 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝐿𝑗) ∈ (Base‘𝑌))
10159adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿𝑗)) = (𝑗 · 𝑋))
102101eqcomd 2741 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑗 ∈ ℤ) → (𝑗 · 𝑋) = (𝐹‘(𝐿𝑗)))
103 fveq2 6907 . . . . . . . . . . . 12 (𝑎 = (𝐿𝑗) → (𝐹𝑎) = (𝐹‘(𝐿𝑗)))
104103rspceeqv 3645 . . . . . . . . . . 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 2739 . . . . . . . . . . 11 (𝑧 = (𝑗 · 𝑋) → (𝑧 = (𝐹𝑎) ↔ (𝑗 · 𝑋) = (𝐹𝑎)))
107106rexbidv 3177 . . . . . . . . . 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 3153 . . . . . . . 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 3165 . . . . . 6 (𝜑 → (∀𝑧𝐵𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∀𝑧𝐵𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎)))
11293, 111mpd 15 . . . . 5 (𝜑 → ∀𝑧𝐵𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹𝑎))
113 dffo3 7122 . . . . 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 6570 . . . 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 19293 . . 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 19302 . 2 (𝐹 ∈ (𝑌 GrpIso 𝐺) → 𝑌𝑔 𝐺)
120 gicsym 19306 . 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 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  ifcif 4531  cop 4637   class class class wbr 5148  cmpt 5231  ran crn 5690  wf 6559  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Fincfn 8984  0cc0 11153   + caddc 11156  0cn0 12524  cz 12611  chash 14366  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  .gcmg 19098   GrpHom cghm 19243   GrpIso cgim 19288  𝑔 cgic 19289  CycGrpccyg 19910  Ringcrg 20251  CRingccrg 20252   RingHom crh 20486  ringczring 21475  ℤRHomczrh 21528  ℤ/nczn 21531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232  ax-mulf 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-acn 9980  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-rp 13033  df-fz 13545  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-dvds 16288  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  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 17488  df-imas 17555  df-qus 17556  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-nsg 19155  df-eqg 19156  df-ghm 19244  df-gim 19290  df-gic 19291  df-od 19561  df-cmn 19815  df-abl 19816  df-cyg 19911  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-rhm 20489  df-subrng 20563  df-subrg 20587  df-lmod 20877  df-lss 20948  df-lsp 20988  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-2idl 21278  df-cnfld 21383  df-zring 21476  df-zrh 21532  df-zn 21535
This theorem is referenced by:  cygzn  21607
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