Step | Hyp | Ref
| Expression |
1 | | eqid 2739 |
. . . 4
⊢
(Base‘𝑌) =
(Base‘𝑌) |
2 | | cygzn.b |
. . . 4
⊢ 𝐵 = (Base‘𝐺) |
3 | | eqid 2739 |
. . . 4
⊢
(+g‘𝑌) = (+g‘𝑌) |
4 | | eqid 2739 |
. . . 4
⊢
(+g‘𝐺) = (+g‘𝐺) |
5 | | cygzn.n |
. . . . . 6
⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0) |
6 | | hashcl 13822 |
. . . . . . . 8
⊢ (𝐵 ∈ Fin →
(♯‘𝐵) ∈
ℕ0) |
7 | 6 | adantl 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 ∈ Fin) → (♯‘𝐵) ∈
ℕ0) |
8 | | 0nn0 12004 |
. . . . . . . 8
⊢ 0 ∈
ℕ0 |
9 | 8 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐵 ∈ Fin) → 0 ∈
ℕ0) |
10 | 7, 9 | ifclda 4459 |
. . . . . 6
⊢ (𝜑 → if(𝐵 ∈ Fin, (♯‘𝐵), 0) ∈
ℕ0) |
11 | 5, 10 | eqeltrid 2838 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
12 | | cygzn.y |
. . . . . 6
⊢ 𝑌 =
(ℤ/nℤ‘𝑁) |
13 | 12 | zncrng 20376 |
. . . . 5
⊢ (𝑁 ∈ ℕ0
→ 𝑌 ∈
CRing) |
14 | | crngring 19441 |
. . . . 5
⊢ (𝑌 ∈ CRing → 𝑌 ∈ Ring) |
15 | | ringgrp 19434 |
. . . . 5
⊢ (𝑌 ∈ Ring → 𝑌 ∈ Grp) |
16 | 11, 13, 14, 15 | 4syl 19 |
. . . 4
⊢ (𝜑 → 𝑌 ∈ Grp) |
17 | | cygzn.g |
. . . . 5
⊢ (𝜑 → 𝐺 ∈ CycGrp) |
18 | | cyggrp 19141 |
. . . . 5
⊢ (𝐺 ∈ CycGrp → 𝐺 ∈ Grp) |
19 | 17, 18 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐺 ∈ Grp) |
20 | | cygzn.m |
. . . . 5
⊢ · =
(.g‘𝐺) |
21 | | cygzn.l |
. . . . 5
⊢ 𝐿 = (ℤRHom‘𝑌) |
22 | | cygzn.e |
. . . . 5
⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵} |
23 | | cygzn.x |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ 𝐸) |
24 | | cygzn.f |
. . . . 5
⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ 〈(𝐿‘𝑚), (𝑚 · 𝑋)〉) |
25 | 2, 5, 12, 20, 21, 22, 17, 23, 24 | cygznlem2a 20399 |
. . . 4
⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵) |
26 | 12, 1, 21 | znzrhfo 20379 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ0
→ 𝐿:ℤ–onto→(Base‘𝑌)) |
27 | 11, 26 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐿:ℤ–onto→(Base‘𝑌)) |
28 | | foelrn 6895 |
. . . . . . 7
⊢ ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖)) |
29 | 27, 28 | sylan 583 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖)) |
30 | | foelrn 6895 |
. . . . . . 7
⊢ ((𝐿:ℤ–onto→(Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)) |
31 | 27, 30 | sylan 583 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ (Base‘𝑌)) → ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)) |
32 | 29, 31 | anim12dan 622 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))) |
33 | | reeanv 3271 |
. . . . . . 7
⊢
(∃𝑖 ∈
ℤ ∃𝑗 ∈
ℤ (𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) ↔ (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))) |
34 | 19 | adantr 484 |
. . . . . . . . . . 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) |
37 | 2, 20, 22 | iscyggen 19131 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑋)) = 𝐵)) |
38 | 37 | simplbi 501 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∈ 𝐸 → 𝑋 ∈ 𝐵) |
39 | 23, 38 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
40 | 39 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝑋 ∈ 𝐵) |
41 | 2, 20, 4 | mulgdir 18390 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑋 ∈ 𝐵)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g‘𝐺)(𝑗 · 𝑋))) |
42 | 34, 35, 36, 40, 41 | syl13anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑖 + 𝑗) · 𝑋) = ((𝑖 · 𝑋)(+g‘𝐺)(𝑗 · 𝑋))) |
43 | 11, 13 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑌 ∈ CRing) |
44 | 21 | zrhrhm 20345 |
. . . . . . . . . . . . . . 15
⊢ (𝑌 ∈ Ring → 𝐿 ∈ (ℤring
RingHom 𝑌)) |
45 | | rhmghm 19612 |
. . . . . . . . . . . . . . 15
⊢ (𝐿 ∈ (ℤring
RingHom 𝑌) → 𝐿 ∈ (ℤring
GrpHom 𝑌)) |
46 | 43, 14, 44, 45 | 4syl 19 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐿 ∈ (ℤring GrpHom 𝑌)) |
47 | 46 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → 𝐿 ∈ (ℤring GrpHom 𝑌)) |
48 | | zringbas 20308 |
. . . . . . . . . . . . . 14
⊢ ℤ =
(Base‘ℤring) |
49 | | zringplusg 20309 |
. . . . . . . . . . . . . 14
⊢ + =
(+g‘ℤring) |
50 | 48, 49, 3 | ghmlin 18494 |
. . . . . . . . . . . . 13
⊢ ((𝐿 ∈ (ℤring
GrpHom 𝑌) ∧ 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))) |
51 | 47, 35, 36, 50 | syl3anc 1372 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐿‘(𝑖 + 𝑗)) = ((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))) |
52 | 51 | fveq2d 6691 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗)))) |
53 | | zaddcl 12116 |
. . . . . . . . . . . 12
⊢ ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 + 𝑗) ∈ ℤ) |
54 | 2, 5, 12, 20, 21, 22, 17, 23, 24 | cygznlem2 20400 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 + 𝑗) ∈ ℤ) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋)) |
55 | 53, 54 | sylan2 596 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘(𝑖 + 𝑗))) = ((𝑖 + 𝑗) · 𝑋)) |
56 | 52, 55 | eqtr3d 2776 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))) = ((𝑖 + 𝑗) · 𝑋)) |
57 | 2, 5, 12, 20, 21, 22, 17, 23, 24 | cygznlem2 20400 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ ℤ) → (𝐹‘(𝐿‘𝑖)) = (𝑖 · 𝑋)) |
58 | 57 | adantrr 717 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘𝑖)) = (𝑖 · 𝑋)) |
59 | 2, 5, 12, 20, 21, 22, 17, 23, 24 | cygznlem2 20400 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋)) |
60 | 59 | adantrl 716 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋)) |
61 | 58, 60 | oveq12d 7201 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿‘𝑖))(+g‘𝐺)(𝐹‘(𝐿‘𝑗))) = ((𝑖 · 𝑋)(+g‘𝐺)(𝑗 · 𝑋))) |
62 | 42, 56, 61 | 3eqtr4d 2784 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))) = ((𝐹‘(𝐿‘𝑖))(+g‘𝐺)(𝐹‘(𝐿‘𝑗)))) |
63 | | oveq12 7192 |
. . . . . . . . . . 11
⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝑎(+g‘𝑌)𝑏) = ((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))) |
64 | 63 | fveq2d 6691 |
. . . . . . . . . 10
⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗)))) |
65 | | fveq2 6687 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝐿‘𝑖) → (𝐹‘𝑎) = (𝐹‘(𝐿‘𝑖))) |
66 | | fveq2 6687 |
. . . . . . . . . . 11
⊢ (𝑏 = (𝐿‘𝑗) → (𝐹‘𝑏) = (𝐹‘(𝐿‘𝑗))) |
67 | 65, 66 | oveqan12d 7202 |
. . . . . . . . . 10
⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏)) = ((𝐹‘(𝐿‘𝑖))(+g‘𝐺)(𝐹‘(𝐿‘𝑗)))) |
68 | 64, 67 | eqeq12d 2755 |
. . . . . . . . 9
⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘(𝑎(+g‘𝑌)𝑏)) = ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏)) ↔ (𝐹‘((𝐿‘𝑖)(+g‘𝑌)(𝐿‘𝑗))) = ((𝐹‘(𝐿‘𝑖))(+g‘𝐺)(𝐹‘(𝐿‘𝑗))))) |
69 | 62, 68 | syl5ibrcom 250 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏)))) |
70 | 69 | rexlimdvva 3205 |
. . . . . . 7
⊢ (𝜑 → (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏)))) |
71 | 33, 70 | syl5bir 246 |
. . . . . 6
⊢ (𝜑 → ((∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏)))) |
72 | 71 | imp 410 |
. . . . 5
⊢ ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏))) |
73 | 32, 72 | syldan 594 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝐹‘(𝑎(+g‘𝑌)𝑏)) = ((𝐹‘𝑎)(+g‘𝐺)(𝐹‘𝑏))) |
74 | 1, 2, 3, 4, 16, 19, 25, 73 | isghmd 18498 |
. . 3
⊢ (𝜑 → 𝐹 ∈ (𝑌 GrpHom 𝐺)) |
75 | 58, 60 | eqeq12d 2755 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗)) ↔ (𝑖 · 𝑋) = (𝑗 · 𝑋))) |
76 | 2, 5, 12, 20, 21, 22, 17, 23 | cygznlem1 20398 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐿‘𝑖) = (𝐿‘𝑗) ↔ (𝑖 · 𝑋) = (𝑗 · 𝑋))) |
77 | 75, 76 | bitr4d 285 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗)) ↔ (𝐿‘𝑖) = (𝐿‘𝑗))) |
78 | 77 | biimpd 232 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗)) → (𝐿‘𝑖) = (𝐿‘𝑗))) |
79 | 65, 66 | eqeqan12d 2756 |
. . . . . . . . . . . 12
⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ (𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗)))) |
80 | | eqeq12 2753 |
. . . . . . . . . . . 12
⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (𝑎 = 𝑏 ↔ (𝐿‘𝑖) = (𝐿‘𝑗))) |
81 | 79, 80 | imbi12d 348 |
. . . . . . . . . . 11
⊢ ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → (((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏) ↔ ((𝐹‘(𝐿‘𝑖)) = (𝐹‘(𝐿‘𝑗)) → (𝐿‘𝑖) = (𝐿‘𝑗)))) |
82 | 78, 81 | syl5ibrcom 250 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))) |
83 | 82 | rexlimdvva 3205 |
. . . . . . . . 9
⊢ (𝜑 → (∃𝑖 ∈ ℤ ∃𝑗 ∈ ℤ (𝑎 = (𝐿‘𝑖) ∧ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))) |
84 | 33, 83 | syl5bir 246 |
. . . . . . . 8
⊢ (𝜑 → ((∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))) |
85 | 84 | imp 410 |
. . . . . . 7
⊢ ((𝜑 ∧ (∃𝑖 ∈ ℤ 𝑎 = (𝐿‘𝑖) ∧ ∃𝑗 ∈ ℤ 𝑏 = (𝐿‘𝑗))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)) |
86 | 32, 85 | syldan 594 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌))) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)) |
87 | 86 | ralrimivva 3104 |
. . . . 5
⊢ (𝜑 → ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)) |
88 | | dff13 7037 |
. . . . 5
⊢ (𝐹:(Base‘𝑌)–1-1→𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))) |
89 | 25, 87, 88 | sylanbrc 586 |
. . . 4
⊢ (𝜑 → 𝐹:(Base‘𝑌)–1-1→𝐵) |
90 | 2, 20, 22 | iscyggen2 19132 |
. . . . . . . . 9
⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋)))) |
91 | 19, 90 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋)))) |
92 | 23, 91 | mpbid 235 |
. . . . . . 7
⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋))) |
93 | 92 | simprd 499 |
. . . . . 6
⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋)) |
94 | | oveq1 7190 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑗 → (𝑛 · 𝑋) = (𝑗 · 𝑋)) |
95 | 94 | eqeq2d 2750 |
. . . . . . . . 9
⊢ (𝑛 = 𝑗 → (𝑧 = (𝑛 · 𝑋) ↔ 𝑧 = (𝑗 · 𝑋))) |
96 | 95 | cbvrexvw 3351 |
. . . . . . . 8
⊢
(∃𝑛 ∈
ℤ 𝑧 = (𝑛 · 𝑋) ↔ ∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋)) |
97 | 27 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐿:ℤ–onto→(Base‘𝑌)) |
98 | | fof 6603 |
. . . . . . . . . . . . 13
⊢ (𝐿:ℤ–onto→(Base‘𝑌) → 𝐿:ℤ⟶(Base‘𝑌)) |
99 | 97, 98 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐿:ℤ⟶(Base‘𝑌)) |
100 | 99 | ffvelrnda 6874 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝐿‘𝑗) ∈ (Base‘𝑌)) |
101 | 59 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝐹‘(𝐿‘𝑗)) = (𝑗 · 𝑋)) |
102 | 101 | eqcomd 2745 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝑗 · 𝑋) = (𝐹‘(𝐿‘𝑗))) |
103 | | fveq2 6687 |
. . . . . . . . . . . 12
⊢ (𝑎 = (𝐿‘𝑗) → (𝐹‘𝑎) = (𝐹‘(𝐿‘𝑗))) |
104 | 103 | rspceeqv 3544 |
. . . . . . . . . . 11
⊢ (((𝐿‘𝑗) ∈ (Base‘𝑌) ∧ (𝑗 · 𝑋) = (𝐹‘(𝐿‘𝑗))) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎)) |
105 | 100, 102,
104 | syl2anc 587 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎)) |
106 | | eqeq1 2743 |
. . . . . . . . . . 11
⊢ (𝑧 = (𝑗 · 𝑋) → (𝑧 = (𝐹‘𝑎) ↔ (𝑗 · 𝑋) = (𝐹‘𝑎))) |
107 | 106 | rexbidv 3208 |
. . . . . . . . . 10
⊢ (𝑧 = (𝑗 · 𝑋) → (∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎) ↔ ∃𝑎 ∈ (Base‘𝑌)(𝑗 · 𝑋) = (𝐹‘𝑎))) |
108 | 105, 107 | syl5ibrcom 250 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑗 ∈ ℤ) → (𝑧 = (𝑗 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎))) |
109 | 108 | rexlimdva 3195 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (∃𝑗 ∈ ℤ 𝑧 = (𝑗 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎))) |
110 | 96, 109 | syl5bi 245 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎))) |
111 | 110 | ralimdva 3092 |
. . . . . 6
⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ∃𝑛 ∈ ℤ 𝑧 = (𝑛 · 𝑋) → ∀𝑧 ∈ 𝐵 ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎))) |
112 | 93, 111 | mpd 15 |
. . . . 5
⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎)) |
113 | | dffo3 6891 |
. . . . 5
⊢ (𝐹:(Base‘𝑌)–onto→𝐵 ↔ (𝐹:(Base‘𝑌)⟶𝐵 ∧ ∀𝑧 ∈ 𝐵 ∃𝑎 ∈ (Base‘𝑌)𝑧 = (𝐹‘𝑎))) |
114 | 25, 112, 113 | sylanbrc 586 |
. . . 4
⊢ (𝜑 → 𝐹:(Base‘𝑌)–onto→𝐵) |
115 | | df-f1o 6357 |
. . . 4
⊢ (𝐹:(Base‘𝑌)–1-1-onto→𝐵 ↔ (𝐹:(Base‘𝑌)–1-1→𝐵 ∧ 𝐹:(Base‘𝑌)–onto→𝐵)) |
116 | 89, 114, 115 | sylanbrc 586 |
. . 3
⊢ (𝜑 → 𝐹:(Base‘𝑌)–1-1-onto→𝐵) |
117 | 1, 2 | isgim 18533 |
. . 3
⊢ (𝐹 ∈ (𝑌 GrpIso 𝐺) ↔ (𝐹 ∈ (𝑌 GrpHom 𝐺) ∧ 𝐹:(Base‘𝑌)–1-1-onto→𝐵)) |
118 | 74, 116, 117 | sylanbrc 586 |
. 2
⊢ (𝜑 → 𝐹 ∈ (𝑌 GrpIso 𝐺)) |
119 | | brgici 18541 |
. 2
⊢ (𝐹 ∈ (𝑌 GrpIso 𝐺) → 𝑌 ≃𝑔 𝐺) |
120 | | gicsym 18545 |
. 2
⊢ (𝑌 ≃𝑔
𝐺 → 𝐺 ≃𝑔 𝑌) |
121 | 118, 119,
120 | 3syl 18 |
1
⊢ (𝜑 → 𝐺 ≃𝑔 𝑌) |