| Step | Hyp | Ref
| Expression |
| 1 | | prmgrpsimpgd.1 |
. 2
⊢ 𝐵 = (Base‘𝐺) |
| 2 | | eqid 2737 |
. 2
⊢
(0g‘𝐺) = (0g‘𝐺) |
| 3 | | prmgrpsimpgd.2 |
. 2
⊢ (𝜑 → 𝐺 ∈ Grp) |
| 4 | | fveq2 6906 |
. . . . . 6
⊢
({(0g‘𝐺)} = 𝐵 →
(♯‘{(0g‘𝐺)}) = (♯‘𝐵)) |
| 5 | 4 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ {(0g‘𝐺)} = 𝐵) →
(♯‘{(0g‘𝐺)}) = (♯‘𝐵)) |
| 6 | 2 | fvexi 6920 |
. . . . . 6
⊢
(0g‘𝐺) ∈ V |
| 7 | | hashsng 14408 |
. . . . . 6
⊢
((0g‘𝐺) ∈ V →
(♯‘{(0g‘𝐺)}) = 1) |
| 8 | 6, 7 | mp1i 13 |
. . . . 5
⊢ ((𝜑 ∧ {(0g‘𝐺)} = 𝐵) →
(♯‘{(0g‘𝐺)}) = 1) |
| 9 | 5, 8 | eqtr3d 2779 |
. . . 4
⊢ ((𝜑 ∧ {(0g‘𝐺)} = 𝐵) → (♯‘𝐵) = 1) |
| 10 | | prmgrpsimpgd.3 |
. . . . 5
⊢ (𝜑 → (♯‘𝐵) ∈
ℙ) |
| 11 | 10 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ {(0g‘𝐺)} = 𝐵) → (♯‘𝐵) ∈ ℙ) |
| 12 | 9, 11 | eqeltrrd 2842 |
. . 3
⊢ ((𝜑 ∧ {(0g‘𝐺)} = 𝐵) → 1 ∈ ℙ) |
| 13 | | 1nprm 16716 |
. . . 4
⊢ ¬ 1
∈ ℙ |
| 14 | 13 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ {(0g‘𝐺)} = 𝐵) → ¬ 1 ∈
ℙ) |
| 15 | 12, 14 | pm2.65da 817 |
. 2
⊢ (𝜑 → ¬
{(0g‘𝐺)} =
𝐵) |
| 16 | | nsgsubg 19176 |
. . 3
⊢ (𝑥 ∈ (NrmSGrp‘𝐺) → 𝑥 ∈ (SubGrp‘𝐺)) |
| 17 | | eqid 2737 |
. . . . . . . 8
⊢
(♯‘𝐵) =
(♯‘𝐵) |
| 18 | 1 | fvexi 6920 |
. . . . . . . . . 10
⊢ 𝐵 ∈ V |
| 19 | 18 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ V) |
| 20 | | prmnn 16711 |
. . . . . . . . . . 11
⊢
((♯‘𝐵)
∈ ℙ → (♯‘𝐵) ∈ ℕ) |
| 21 | 10, 20 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (♯‘𝐵) ∈
ℕ) |
| 22 | 21 | nnnn0d 12587 |
. . . . . . . . 9
⊢ (𝜑 → (♯‘𝐵) ∈
ℕ0) |
| 23 | | hashvnfin 14399 |
. . . . . . . . 9
⊢ ((𝐵 ∈ V ∧
(♯‘𝐵) ∈
ℕ0) → ((♯‘𝐵) = (♯‘𝐵) → 𝐵 ∈ Fin)) |
| 24 | 19, 22, 23 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → ((♯‘𝐵) = (♯‘𝐵) → 𝐵 ∈ Fin)) |
| 25 | 17, 24 | mpi 20 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ Fin) |
| 26 | 25 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝐵)) → 𝐵 ∈ Fin) |
| 27 | 1 | subgss 19145 |
. . . . . . 7
⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ 𝐵) |
| 28 | 27 | ad2antlr 727 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝐵)) → 𝑥 ⊆ 𝐵) |
| 29 | | simpr 484 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝐵)) → (♯‘𝑥) = (♯‘𝐵)) |
| 30 | 26, 28, 29 | phphashrd 14506 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝐵)) → 𝑥 = 𝐵) |
| 31 | 30 | olcd 875 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ (♯‘𝑥) = (♯‘𝐵)) → (𝑥 = {(0g‘𝐺)} ∨ 𝑥 = 𝐵)) |
| 32 | | simpr 484 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ (♯‘𝑥) = 1) → (♯‘𝑥) = 1) |
| 33 | 2 | subg0cl 19152 |
. . . . . . 7
⊢ (𝑥 ∈ (SubGrp‘𝐺) →
(0g‘𝐺)
∈ 𝑥) |
| 34 | 33 | ad2antlr 727 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ (♯‘𝑥) = 1) → (0g‘𝐺) ∈ 𝑥) |
| 35 | | vex 3484 |
. . . . . . 7
⊢ 𝑥 ∈ V |
| 36 | 35 | a1i 11 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ (♯‘𝑥) = 1) → 𝑥 ∈ V) |
| 37 | 32, 34, 36 | hash1elsn 14410 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ (♯‘𝑥) = 1) → 𝑥 = {(0g‘𝐺)}) |
| 38 | 37 | orcd 874 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) ∧ (♯‘𝑥) = 1) → (𝑥 = {(0g‘𝐺)} ∨ 𝑥 = 𝐵)) |
| 39 | 1 | lagsubg 19213 |
. . . . . . 7
⊢ ((𝑥 ∈ (SubGrp‘𝐺) ∧ 𝐵 ∈ Fin) → (♯‘𝑥) ∥ (♯‘𝐵)) |
| 40 | 25, 39 | sylan2 593 |
. . . . . 6
⊢ ((𝑥 ∈ (SubGrp‘𝐺) ∧ 𝜑) → (♯‘𝑥) ∥ (♯‘𝐵)) |
| 41 | 40 | ancoms 458 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → (♯‘𝑥) ∥ (♯‘𝐵)) |
| 42 | 10 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → (♯‘𝐵) ∈ ℙ) |
| 43 | 25 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → 𝐵 ∈ Fin) |
| 44 | 27 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → 𝑥 ⊆ 𝐵) |
| 45 | 43, 44 | ssfid 9301 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → 𝑥 ∈ Fin) |
| 46 | | hashcl 14395 |
. . . . . . . 8
⊢ (𝑥 ∈ Fin →
(♯‘𝑥) ∈
ℕ0) |
| 47 | 45, 46 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → (♯‘𝑥) ∈
ℕ0) |
| 48 | 33 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → (0g‘𝐺) ∈ 𝑥) |
| 49 | 35 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → 𝑥 ∈ V) |
| 50 | 48, 49 | hashelne0d 14407 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → ¬ (♯‘𝑥) = 0) |
| 51 | 50 | neqned 2947 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → (♯‘𝑥) ≠ 0) |
| 52 | | elnnne0 12540 |
. . . . . . 7
⊢
((♯‘𝑥)
∈ ℕ ↔ ((♯‘𝑥) ∈ ℕ0 ∧
(♯‘𝑥) ≠
0)) |
| 53 | 47, 51, 52 | sylanbrc 583 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → (♯‘𝑥) ∈ ℕ) |
| 54 | | dvdsprime 16724 |
. . . . . 6
⊢
(((♯‘𝐵)
∈ ℙ ∧ (♯‘𝑥) ∈ ℕ) →
((♯‘𝑥) ∥
(♯‘𝐵) ↔
((♯‘𝑥) =
(♯‘𝐵) ∨
(♯‘𝑥) =
1))) |
| 55 | 42, 53, 54 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → ((♯‘𝑥) ∥ (♯‘𝐵) ↔ ((♯‘𝑥) = (♯‘𝐵) ∨ (♯‘𝑥) = 1))) |
| 56 | 41, 55 | mpbid 232 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → ((♯‘𝑥) = (♯‘𝐵) ∨ (♯‘𝑥) = 1)) |
| 57 | 31, 38, 56 | mpjaodan 961 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (SubGrp‘𝐺)) → (𝑥 = {(0g‘𝐺)} ∨ 𝑥 = 𝐵)) |
| 58 | 16, 57 | sylan2 593 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ (NrmSGrp‘𝐺)) → (𝑥 = {(0g‘𝐺)} ∨ 𝑥 = 𝐵)) |
| 59 | 1, 2, 3, 15, 58 | 2nsgsimpgd 20122 |
1
⊢ (𝜑 → 𝐺 ∈ SimpGrp) |