Step | Hyp | Ref
| Expression |
1 | | sylow2a.x |
. . 3
⊢ 𝑋 = (Base‘𝐺) |
2 | | sylow2a.m |
. . 3
⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑌)) |
3 | | sylow2a.p |
. . 3
⊢ (𝜑 → 𝑃 pGrp 𝐺) |
4 | | sylow2a.f |
. . 3
⊢ (𝜑 → 𝑋 ∈ Fin) |
5 | | sylow2a.y |
. . 3
⊢ (𝜑 → 𝑌 ∈ Fin) |
6 | | sylow2a.z |
. . 3
⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢} |
7 | | sylow2a.r |
. . 3
⊢ ∼ =
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} |
8 | 1, 2, 3, 4, 5, 6, 7 | sylow2alem2 19232 |
. 2
⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧)) |
9 | | inass 4154 |
. . . . . . 7
⊢ (((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)) = ((𝑌 / ∼ ) ∩ (𝒫
𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫
𝑍))) |
10 | | disjdif 4406 |
. . . . . . . 8
⊢
(𝒫 𝑍 ∩
((𝑌 / ∼ )
∖ 𝒫 𝑍)) =
∅ |
11 | 10 | ineq2i 4144 |
. . . . . . 7
⊢ ((𝑌 / ∼ ) ∩ (𝒫
𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫
𝑍))) = ((𝑌 / ∼ ) ∩
∅) |
12 | | in0 4326 |
. . . . . . 7
⊢ ((𝑌 / ∼ ) ∩ ∅) =
∅ |
13 | 9, 11, 12 | 3eqtri 2771 |
. . . . . 6
⊢ (((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)) =
∅ |
14 | 13 | a1i 11 |
. . . . 5
⊢ (𝜑 → (((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)) =
∅) |
15 | | inundif 4413 |
. . . . . . 7
⊢ (((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)) = (𝑌 / ∼ ) |
16 | 15 | eqcomi 2748 |
. . . . . 6
⊢ (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)) |
17 | 16 | a1i 11 |
. . . . 5
⊢ (𝜑 → (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫
𝑍))) |
18 | | pwfi 8970 |
. . . . . . 7
⊢ (𝑌 ∈ Fin ↔ 𝒫
𝑌 ∈
Fin) |
19 | 5, 18 | sylib 217 |
. . . . . 6
⊢ (𝜑 → 𝒫 𝑌 ∈ Fin) |
20 | 7, 1 | gaorber 18923 |
. . . . . . . 8
⊢ ( ⊕ ∈
(𝐺 GrpAct 𝑌) → ∼ Er 𝑌) |
21 | 2, 20 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ∼ Er 𝑌) |
22 | 21 | qsss 8576 |
. . . . . 6
⊢ (𝜑 → (𝑌 / ∼ ) ⊆ 𝒫
𝑌) |
23 | 19, 22 | ssfid 9051 |
. . . . 5
⊢ (𝜑 → (𝑌 / ∼ ) ∈
Fin) |
24 | 5 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑌 ∈ Fin) |
25 | 22 | sselda 3922 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ 𝒫 𝑌) |
26 | 25 | elpwid 4545 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ⊆ 𝑌) |
27 | 24, 26 | ssfid 9051 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ Fin) |
28 | | hashcl 14080 |
. . . . . . 7
⊢ (𝑧 ∈ Fin →
(♯‘𝑧) ∈
ℕ0) |
29 | 27, 28 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) →
(♯‘𝑧) ∈
ℕ0) |
30 | 29 | nn0cnd 12304 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) →
(♯‘𝑧) ∈
ℂ) |
31 | 14, 17, 23, 30 | fsumsplit 15462 |
. . . 4
⊢ (𝜑 → Σ𝑧 ∈ (𝑌 / ∼
)(♯‘𝑧) =
(Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)(♯‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧))) |
32 | 21, 5 | qshash 15548 |
. . . 4
⊢ (𝜑 → (♯‘𝑌) = Σ𝑧 ∈ (𝑌 / ∼
)(♯‘𝑧)) |
33 | | inss1 4163 |
. . . . . . . 8
⊢ ((𝑌 / ∼ ) ∩ 𝒫
𝑍) ⊆ (𝑌 / ∼ ) |
34 | | ssfi 8965 |
. . . . . . . 8
⊢ (((𝑌 / ∼ ) ∈ Fin ∧
((𝑌 / ∼ )
∩ 𝒫 𝑍) ⊆
(𝑌 / ∼ ))
→ ((𝑌 / ∼ )
∩ 𝒫 𝑍) ∈
Fin) |
35 | 23, 33, 34 | sylancl 586 |
. . . . . . 7
⊢ (𝜑 → ((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∈
Fin) |
36 | | ax-1cn 10938 |
. . . . . . 7
⊢ 1 ∈
ℂ |
37 | | fsumconst 15511 |
. . . . . . 7
⊢ ((((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∈ Fin ∧ 1
∈ ℂ) → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)1 =
((♯‘((𝑌
/ ∼ ) ∩ 𝒫
𝑍)) ·
1)) |
38 | 35, 36, 37 | sylancl 586 |
. . . . . 6
⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)1 =
((♯‘((𝑌
/ ∼ ) ∩ 𝒫
𝑍)) ·
1)) |
39 | | elin 3904 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍) ↔ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍)) |
40 | | eqid 2739 |
. . . . . . . . . . . . 13
⊢ (𝑌 / ∼ ) = (𝑌 / ∼ ) |
41 | | sseq1 3947 |
. . . . . . . . . . . . . . 15
⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ⊆ 𝑍)) |
42 | | velpw 4539 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ 𝒫 𝑍 ↔ 𝑧 ⊆ 𝑍) |
43 | 41, 42 | bitr4di 289 |
. . . . . . . . . . . . . 14
⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ∈ 𝒫 𝑍)) |
44 | | breq1 5078 |
. . . . . . . . . . . . . 14
⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ≈
1o ↔ 𝑧
≈ 1o)) |
45 | 43, 44 | imbi12d 345 |
. . . . . . . . . . . . 13
⊢ ([𝑤] ∼ = 𝑧 → (([𝑤] ∼ ⊆ 𝑍 → [𝑤] ∼ ≈
1o) ↔ (𝑧
∈ 𝒫 𝑍 →
𝑧 ≈
1o))) |
46 | 21 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∼ Er 𝑌) |
47 | | simpr 485 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌) |
48 | 46, 47 | erref 8527 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∼ 𝑤) |
49 | | vex 3437 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑤 ∈ V |
50 | 49, 49 | elec 8551 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ∈ [𝑤] ∼ ↔ 𝑤 ∼ 𝑤) |
51 | 48, 50 | sylibr 233 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ [𝑤] ∼ ) |
52 | | ssel 3915 |
. . . . . . . . . . . . . . 15
⊢ ([𝑤] ∼ ⊆ 𝑍 → (𝑤 ∈ [𝑤] ∼ → 𝑤 ∈ 𝑍)) |
53 | 51, 52 | syl5com 31 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ([𝑤] ∼ ⊆ 𝑍 → 𝑤 ∈ 𝑍)) |
54 | 1, 2, 3, 4, 5, 6, 7 | sylow2alem1 19231 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ = {𝑤}) |
55 | 49 | ensn1 8816 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑤} ≈
1o |
56 | 54, 55 | eqbrtrdi 5114 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ≈
1o) |
57 | 56 | ex 413 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑤 ∈ 𝑍 → [𝑤] ∼ ≈
1o)) |
58 | 57 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑤 ∈ 𝑍 → [𝑤] ∼ ≈
1o)) |
59 | 53, 58 | syld 47 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ([𝑤] ∼ ⊆ 𝑍 → [𝑤] ∼ ≈
1o)) |
60 | 40, 45, 59 | ectocld 8582 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (𝑧 ∈ 𝒫 𝑍 → 𝑧 ≈ 1o)) |
61 | 60 | impr 455 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍)) → 𝑧 ≈ 1o) |
62 | 39, 61 | sylan2b 594 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → 𝑧 ≈
1o) |
63 | | en1b 8822 |
. . . . . . . . . 10
⊢ (𝑧 ≈ 1o ↔
𝑧 = {∪ 𝑧}) |
64 | 62, 63 | sylib 217 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → 𝑧 = {∪
𝑧}) |
65 | 64 | fveq2d 6787 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) →
(♯‘𝑧) =
(♯‘{∪ 𝑧})) |
66 | | vuniex 7601 |
. . . . . . . . 9
⊢ ∪ 𝑧
∈ V |
67 | | hashsng 14093 |
. . . . . . . . 9
⊢ (∪ 𝑧
∈ V → (♯‘{∪ 𝑧}) = 1) |
68 | 66, 67 | ax-mp 5 |
. . . . . . . 8
⊢
(♯‘{∪ 𝑧}) = 1 |
69 | 65, 68 | eqtrdi 2795 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) →
(♯‘𝑧) =
1) |
70 | 69 | sumeq2dv 15424 |
. . . . . 6
⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)(♯‘𝑧) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)1) |
71 | 6 | ssrab3 4016 |
. . . . . . . . . . 11
⊢ 𝑍 ⊆ 𝑌 |
72 | | ssfi 8965 |
. . . . . . . . . . 11
⊢ ((𝑌 ∈ Fin ∧ 𝑍 ⊆ 𝑌) → 𝑍 ∈ Fin) |
73 | 5, 71, 72 | sylancl 586 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑍 ∈ Fin) |
74 | | hashcl 14080 |
. . . . . . . . . 10
⊢ (𝑍 ∈ Fin →
(♯‘𝑍) ∈
ℕ0) |
75 | 73, 74 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (♯‘𝑍) ∈
ℕ0) |
76 | 75 | nn0cnd 12304 |
. . . . . . . 8
⊢ (𝜑 → (♯‘𝑍) ∈
ℂ) |
77 | 76 | mulid1d 11001 |
. . . . . . 7
⊢ (𝜑 → ((♯‘𝑍) · 1) =
(♯‘𝑍)) |
78 | 6, 5 | rabexd 5258 |
. . . . . . . . 9
⊢ (𝜑 → 𝑍 ∈ V) |
79 | | eqid 2739 |
. . . . . . . . . 10
⊢ (𝑤 ∈ 𝑍 ↦ {𝑤}) = (𝑤 ∈ 𝑍 ↦ {𝑤}) |
80 | 7 | relopabiv 5732 |
. . . . . . . . . . . . . . 15
⊢ Rel ∼ |
81 | | relssdmrn 6176 |
. . . . . . . . . . . . . . 15
⊢ (Rel
∼
→ ∼ ⊆ (dom ∼
× ran ∼ )) |
82 | 80, 81 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ∼
⊆ (dom ∼ × ran ∼
) |
83 | | erdm 8517 |
. . . . . . . . . . . . . . . . 17
⊢ ( ∼ Er
𝑌 → dom ∼ =
𝑌) |
84 | 21, 83 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → dom ∼ = 𝑌) |
85 | 84, 5 | eqeltrd 2840 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → dom ∼ ∈
Fin) |
86 | | errn 8529 |
. . . . . . . . . . . . . . . . 17
⊢ ( ∼ Er
𝑌 → ran ∼ =
𝑌) |
87 | 21, 86 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ran ∼ = 𝑌) |
88 | 87, 5 | eqeltrd 2840 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ran ∼ ∈
Fin) |
89 | 85, 88 | xpexd 7610 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (dom ∼ × ran ∼ )
∈ V) |
90 | | ssexg 5248 |
. . . . . . . . . . . . . 14
⊢ (( ∼
⊆ (dom ∼ × ran ∼ )
∧ (dom ∼ × ran ∼ )
∈ V) → ∼ ∈
V) |
91 | 82, 89, 90 | sylancr 587 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∼ ∈
V) |
92 | | simpr 485 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑍) |
93 | 71, 92 | sselid 3920 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑌) |
94 | | ecelqsg 8570 |
. . . . . . . . . . . . 13
⊢ (( ∼ ∈
V ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ∈ (𝑌 / ∼ )) |
95 | 91, 93, 94 | syl2an2r 682 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ∈ (𝑌 / ∼ )) |
96 | 54, 95 | eqeltrrd 2841 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ (𝑌 / ∼ )) |
97 | | snelpwi 5361 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ 𝑍 → {𝑤} ∈ 𝒫 𝑍) |
98 | 97 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ 𝒫 𝑍) |
99 | 96, 98 | elind 4129 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) |
100 | | simpr 485 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) |
101 | 100 | elin2d 4134 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → 𝑧 ∈ 𝒫 𝑍) |
102 | 101 | elpwid 4545 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → 𝑧 ⊆ 𝑍) |
103 | 64, 102 | eqsstrrd 3961 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → {∪ 𝑧}
⊆ 𝑍) |
104 | 66 | snss 4720 |
. . . . . . . . . . 11
⊢ (∪ 𝑧
∈ 𝑍 ↔ {∪ 𝑧}
⊆ 𝑍) |
105 | 103, 104 | sylibr 233 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → ∪ 𝑧
∈ 𝑍) |
106 | | sneq 4572 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = ∪
𝑧 → {𝑤} = {∪ 𝑧}) |
107 | 106 | eqeq2d 2750 |
. . . . . . . . . . . . 13
⊢ (𝑤 = ∪
𝑧 → (𝑧 = {𝑤} ↔ 𝑧 = {∪ 𝑧})) |
108 | 64, 107 | syl5ibrcom 246 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → (𝑤 = ∪
𝑧 → 𝑧 = {𝑤})) |
109 | 108 | adantrl 713 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍))) → (𝑤 = ∪
𝑧 → 𝑧 = {𝑤})) |
110 | | unieq 4851 |
. . . . . . . . . . . 12
⊢ (𝑧 = {𝑤} → ∪ 𝑧 = ∪
{𝑤}) |
111 | 49 | unisn 4862 |
. . . . . . . . . . . 12
⊢ ∪ {𝑤}
= 𝑤 |
112 | 110, 111 | eqtr2di 2796 |
. . . . . . . . . . 11
⊢ (𝑧 = {𝑤} → 𝑤 = ∪ 𝑧) |
113 | 109, 112 | impbid1 224 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍))) → (𝑤 = ∪
𝑧 ↔ 𝑧 = {𝑤})) |
114 | 79, 99, 105, 113 | f1o2d 7532 |
. . . . . . . . 9
⊢ (𝜑 → (𝑤 ∈ 𝑍 ↦ {𝑤}):𝑍–1-1-onto→((𝑌 / ∼ ) ∩ 𝒫
𝑍)) |
115 | 78, 114 | hasheqf1od 14077 |
. . . . . . . 8
⊢ (𝜑 → (♯‘𝑍) = (♯‘((𝑌 / ∼ ) ∩ 𝒫
𝑍))) |
116 | 115 | oveq1d 7299 |
. . . . . . 7
⊢ (𝜑 → ((♯‘𝑍) · 1) =
((♯‘((𝑌
/ ∼ ) ∩ 𝒫
𝑍)) ·
1)) |
117 | 77, 116 | eqtr3d 2781 |
. . . . . 6
⊢ (𝜑 → (♯‘𝑍) = ((♯‘((𝑌 / ∼ ) ∩ 𝒫
𝑍)) ·
1)) |
118 | 38, 70, 117 | 3eqtr4rd 2790 |
. . . . 5
⊢ (𝜑 → (♯‘𝑍) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)(♯‘𝑧)) |
119 | 118 | oveq1d 7299 |
. . . 4
⊢ (𝜑 → ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧)) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)(♯‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧))) |
120 | 31, 32, 119 | 3eqtr4rd 2790 |
. . 3
⊢ (𝜑 → ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧)) = (♯‘𝑌)) |
121 | | hashcl 14080 |
. . . . . 6
⊢ (𝑌 ∈ Fin →
(♯‘𝑌) ∈
ℕ0) |
122 | 5, 121 | syl 17 |
. . . . 5
⊢ (𝜑 → (♯‘𝑌) ∈
ℕ0) |
123 | 122 | nn0cnd 12304 |
. . . 4
⊢ (𝜑 → (♯‘𝑌) ∈
ℂ) |
124 | | diffi 8971 |
. . . . . 6
⊢ ((𝑌 / ∼ ) ∈ Fin →
((𝑌 / ∼ )
∖ 𝒫 𝑍) ∈
Fin) |
125 | 23, 124 | syl 17 |
. . . . 5
⊢ (𝜑 → ((𝑌 / ∼ ) ∖ 𝒫
𝑍) ∈
Fin) |
126 | | eldifi 4062 |
. . . . . 6
⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍) → 𝑧 ∈ (𝑌 / ∼ )) |
127 | 126, 30 | sylan2 593 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)) →
(♯‘𝑧) ∈
ℂ) |
128 | 125, 127 | fsumcl 15454 |
. . . 4
⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧) ∈
ℂ) |
129 | 123, 76, 128 | subaddd 11359 |
. . 3
⊢ (𝜑 → (((♯‘𝑌) − (♯‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧) ↔ ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧)) = (♯‘𝑌))) |
130 | 120, 129 | mpbird 256 |
. 2
⊢ (𝜑 → ((♯‘𝑌) − (♯‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧)) |
131 | 8, 130 | breqtrrd 5103 |
1
⊢ (𝜑 → 𝑃 ∥ ((♯‘𝑌) − (♯‘𝑍))) |