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 18745 |
. 2
⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧)) |
9 | | inass 4198 |
. . . . . . 7
⊢ (((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)) = ((𝑌 / ∼ ) ∩ (𝒫
𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫
𝑍))) |
10 | | disjdif 4423 |
. . . . . . . 8
⊢
(𝒫 𝑍 ∩
((𝑌 / ∼ )
∖ 𝒫 𝑍)) =
∅ |
11 | 10 | ineq2i 4188 |
. . . . . . 7
⊢ ((𝑌 / ∼ ) ∩ (𝒫
𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫
𝑍))) = ((𝑌 / ∼ ) ∩
∅) |
12 | | in0 4347 |
. . . . . . 7
⊢ ((𝑌 / ∼ ) ∩ ∅) =
∅ |
13 | 9, 11, 12 | 3eqtri 2850 |
. . . . . 6
⊢ (((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)) =
∅ |
14 | 13 | a1i 11 |
. . . . 5
⊢ (𝜑 → (((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)) =
∅) |
15 | | inundif 4429 |
. . . . . . 7
⊢ (((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)) = (𝑌 / ∼ ) |
16 | 15 | eqcomi 2832 |
. . . . . 6
⊢ (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)) |
17 | 16 | a1i 11 |
. . . . 5
⊢ (𝜑 → (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫
𝑍))) |
18 | | pwfi 8821 |
. . . . . . 7
⊢ (𝑌 ∈ Fin ↔ 𝒫
𝑌 ∈
Fin) |
19 | 5, 18 | sylib 220 |
. . . . . 6
⊢ (𝜑 → 𝒫 𝑌 ∈ Fin) |
20 | 7, 1 | gaorber 18440 |
. . . . . . . 8
⊢ ( ⊕ ∈
(𝐺 GrpAct 𝑌) → ∼ Er 𝑌) |
21 | 2, 20 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ∼ Er 𝑌) |
22 | 21 | qsss 8360 |
. . . . . 6
⊢ (𝜑 → (𝑌 / ∼ ) ⊆ 𝒫
𝑌) |
23 | 19, 22 | ssfid 8743 |
. . . . 5
⊢ (𝜑 → (𝑌 / ∼ ) ∈
Fin) |
24 | 5 | adantr 483 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑌 ∈ Fin) |
25 | 22 | sselda 3969 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ 𝒫 𝑌) |
26 | 25 | elpwid 4552 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ⊆ 𝑌) |
27 | 24, 26 | ssfid 8743 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ Fin) |
28 | | hashcl 13720 |
. . . . . . 7
⊢ (𝑧 ∈ Fin →
(♯‘𝑧) ∈
ℕ0) |
29 | 27, 28 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) →
(♯‘𝑧) ∈
ℕ0) |
30 | 29 | nn0cnd 11960 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) →
(♯‘𝑧) ∈
ℂ) |
31 | 14, 17, 23, 30 | fsumsplit 15099 |
. . . 4
⊢ (𝜑 → Σ𝑧 ∈ (𝑌 / ∼
)(♯‘𝑧) =
(Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)(♯‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧))) |
32 | 21, 5 | qshash 15184 |
. . . 4
⊢ (𝜑 → (♯‘𝑌) = Σ𝑧 ∈ (𝑌 / ∼
)(♯‘𝑧)) |
33 | | inss1 4207 |
. . . . . . . 8
⊢ ((𝑌 / ∼ ) ∩ 𝒫
𝑍) ⊆ (𝑌 / ∼ ) |
34 | | ssfi 8740 |
. . . . . . . 8
⊢ (((𝑌 / ∼ ) ∈ Fin ∧
((𝑌 / ∼ )
∩ 𝒫 𝑍) ⊆
(𝑌 / ∼ ))
→ ((𝑌 / ∼ )
∩ 𝒫 𝑍) ∈
Fin) |
35 | 23, 33, 34 | sylancl 588 |
. . . . . . 7
⊢ (𝜑 → ((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∈
Fin) |
36 | | ax-1cn 10597 |
. . . . . . 7
⊢ 1 ∈
ℂ |
37 | | fsumconst 15147 |
. . . . . . 7
⊢ ((((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∈ Fin ∧ 1
∈ ℂ) → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)1 =
((♯‘((𝑌
/ ∼ ) ∩ 𝒫
𝑍)) ·
1)) |
38 | 35, 36, 37 | sylancl 588 |
. . . . . 6
⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)1 =
((♯‘((𝑌
/ ∼ ) ∩ 𝒫
𝑍)) ·
1)) |
39 | | elin 4171 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍) ↔ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍)) |
40 | | eqid 2823 |
. . . . . . . . . . . . 13
⊢ (𝑌 / ∼ ) = (𝑌 / ∼ ) |
41 | | sseq1 3994 |
. . . . . . . . . . . . . . 15
⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ⊆ 𝑍)) |
42 | | velpw 4546 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ 𝒫 𝑍 ↔ 𝑧 ⊆ 𝑍) |
43 | 41, 42 | syl6bbr 291 |
. . . . . . . . . . . . . 14
⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ∈ 𝒫 𝑍)) |
44 | | breq1 5071 |
. . . . . . . . . . . . . 14
⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ≈
1o ↔ 𝑧
≈ 1o)) |
45 | 43, 44 | imbi12d 347 |
. . . . . . . . . . . . 13
⊢ ([𝑤] ∼ = 𝑧 → (([𝑤] ∼ ⊆ 𝑍 → [𝑤] ∼ ≈
1o) ↔ (𝑧
∈ 𝒫 𝑍 →
𝑧 ≈
1o))) |
46 | 21 | adantr 483 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∼ Er 𝑌) |
47 | | simpr 487 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌) |
48 | 46, 47 | erref 8311 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∼ 𝑤) |
49 | | vex 3499 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑤 ∈ V |
50 | 49, 49 | elec 8335 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ∈ [𝑤] ∼ ↔ 𝑤 ∼ 𝑤) |
51 | 48, 50 | sylibr 236 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ [𝑤] ∼ ) |
52 | | ssel 3963 |
. . . . . . . . . . . . . . 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 18744 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ = {𝑤}) |
55 | 49 | ensn1 8575 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑤} ≈
1o |
56 | 54, 55 | eqbrtrdi 5107 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ≈
1o) |
57 | 56 | ex 415 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑤 ∈ 𝑍 → [𝑤] ∼ ≈
1o)) |
58 | 57 | adantr 483 |
. . . . . . . . . . . . . 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 8366 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (𝑧 ∈ 𝒫 𝑍 → 𝑧 ≈ 1o)) |
61 | 60 | impr 457 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍)) → 𝑧 ≈ 1o) |
62 | 39, 61 | sylan2b 595 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → 𝑧 ≈
1o) |
63 | | en1b 8579 |
. . . . . . . . . 10
⊢ (𝑧 ≈ 1o ↔
𝑧 = {∪ 𝑧}) |
64 | 62, 63 | sylib 220 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → 𝑧 = {∪
𝑧}) |
65 | 64 | fveq2d 6676 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) →
(♯‘𝑧) =
(♯‘{∪ 𝑧})) |
66 | | vuniex 7467 |
. . . . . . . . 9
⊢ ∪ 𝑧
∈ V |
67 | | hashsng 13733 |
. . . . . . . . 9
⊢ (∪ 𝑧
∈ V → (♯‘{∪ 𝑧}) = 1) |
68 | 66, 67 | ax-mp 5 |
. . . . . . . 8
⊢
(♯‘{∪ 𝑧}) = 1 |
69 | 65, 68 | syl6eq 2874 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) →
(♯‘𝑧) =
1) |
70 | 69 | sumeq2dv 15062 |
. . . . . 6
⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)(♯‘𝑧) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)1) |
71 | 6 | ssrab3 4059 |
. . . . . . . . . . 11
⊢ 𝑍 ⊆ 𝑌 |
72 | | ssfi 8740 |
. . . . . . . . . . 11
⊢ ((𝑌 ∈ Fin ∧ 𝑍 ⊆ 𝑌) → 𝑍 ∈ Fin) |
73 | 5, 71, 72 | sylancl 588 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑍 ∈ Fin) |
74 | | hashcl 13720 |
. . . . . . . . . 10
⊢ (𝑍 ∈ Fin →
(♯‘𝑍) ∈
ℕ0) |
75 | 73, 74 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (♯‘𝑍) ∈
ℕ0) |
76 | 75 | nn0cnd 11960 |
. . . . . . . 8
⊢ (𝜑 → (♯‘𝑍) ∈
ℂ) |
77 | 76 | mulid1d 10660 |
. . . . . . 7
⊢ (𝜑 → ((♯‘𝑍) · 1) =
(♯‘𝑍)) |
78 | 6, 5 | rabexd 5238 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑍 ∈ V) |
79 | | inss2 4208 |
. . . . . . . . . . 11
⊢ ((𝑌 / ∼ ) ∩ 𝒫
𝑍) ⊆ 𝒫 𝑍 |
80 | 73 | pwexd 5282 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝒫 𝑍 ∈ V) |
81 | | ssexg 5229 |
. . . . . . . . . . 11
⊢ ((((𝑌 / ∼ ) ∩ 𝒫
𝑍) ⊆ 𝒫 𝑍 ∧ 𝒫 𝑍 ∈ V) → ((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∈
V) |
82 | 79, 80, 81 | sylancr 589 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∈
V) |
83 | 7 | relopabi 5696 |
. . . . . . . . . . . . . . . . 17
⊢ Rel ∼ |
84 | | relssdmrn 6123 |
. . . . . . . . . . . . . . . . 17
⊢ (Rel
∼
→ ∼ ⊆ (dom ∼
× ran ∼ )) |
85 | 83, 84 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ ∼
⊆ (dom ∼ × ran ∼
) |
86 | | erdm 8301 |
. . . . . . . . . . . . . . . . . . 19
⊢ ( ∼ Er
𝑌 → dom ∼ =
𝑌) |
87 | 21, 86 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → dom ∼ = 𝑌) |
88 | 87, 5 | eqeltrd 2915 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → dom ∼ ∈
Fin) |
89 | | errn 8313 |
. . . . . . . . . . . . . . . . . . 19
⊢ ( ∼ Er
𝑌 → ran ∼ =
𝑌) |
90 | 21, 89 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ran ∼ = 𝑌) |
91 | 90, 5 | eqeltrd 2915 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ran ∼ ∈
Fin) |
92 | 88, 91 | xpexd 7476 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (dom ∼ × ran ∼ )
∈ V) |
93 | | ssexg 5229 |
. . . . . . . . . . . . . . . 16
⊢ (( ∼
⊆ (dom ∼ × ran ∼ )
∧ (dom ∼ × ran ∼ )
∈ V) → ∼ ∈
V) |
94 | 85, 92, 93 | sylancr 589 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∼ ∈
V) |
95 | 94 | adantr 483 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ∼ ∈
V) |
96 | | simpr 487 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑍) |
97 | 71, 96 | sseldi 3967 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑌) |
98 | | ecelqsg 8354 |
. . . . . . . . . . . . . 14
⊢ (( ∼ ∈
V ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ∈ (𝑌 / ∼ )) |
99 | 95, 97, 98 | syl2anc 586 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ∈ (𝑌 / ∼ )) |
100 | 54, 99 | eqeltrrd 2916 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ (𝑌 / ∼ )) |
101 | | snelpwi 5339 |
. . . . . . . . . . . . 13
⊢ (𝑤 ∈ 𝑍 → {𝑤} ∈ 𝒫 𝑍) |
102 | 101 | adantl 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ 𝒫 𝑍) |
103 | 100, 102 | elind 4173 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) |
104 | 103 | ex 415 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑤 ∈ 𝑍 → {𝑤} ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍))) |
105 | | simpr 487 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) |
106 | 79, 105 | sseldi 3967 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → 𝑧 ∈ 𝒫 𝑍) |
107 | 106 | elpwid 4552 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → 𝑧 ⊆ 𝑍) |
108 | 64, 107 | eqsstrrd 4008 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → {∪ 𝑧}
⊆ 𝑍) |
109 | 66 | snss 4720 |
. . . . . . . . . . . 12
⊢ (∪ 𝑧
∈ 𝑍 ↔ {∪ 𝑧}
⊆ 𝑍) |
110 | 108, 109 | sylibr 236 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → ∪ 𝑧
∈ 𝑍) |
111 | 110 | ex 415 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍) → ∪ 𝑧
∈ 𝑍)) |
112 | | sneq 4579 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = ∪
𝑧 → {𝑤} = {∪ 𝑧}) |
113 | 112 | eqeq2d 2834 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = ∪
𝑧 → (𝑧 = {𝑤} ↔ 𝑧 = {∪ 𝑧})) |
114 | 64, 113 | syl5ibrcom 249 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → (𝑤 = ∪
𝑧 → 𝑧 = {𝑤})) |
115 | 114 | adantrl 714 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍))) → (𝑤 = ∪
𝑧 → 𝑧 = {𝑤})) |
116 | | unieq 4851 |
. . . . . . . . . . . . 13
⊢ (𝑧 = {𝑤} → ∪ 𝑧 = ∪
{𝑤}) |
117 | 49 | unisn 4860 |
. . . . . . . . . . . . 13
⊢ ∪ {𝑤}
= 𝑤 |
118 | 116, 117 | syl6req 2875 |
. . . . . . . . . . . 12
⊢ (𝑧 = {𝑤} → 𝑤 = ∪ 𝑧) |
119 | 115, 118 | impbid1 227 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍))) → (𝑤 = ∪
𝑧 ↔ 𝑧 = {𝑤})) |
120 | 119 | ex 415 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) → (𝑤 = ∪
𝑧 ↔ 𝑧 = {𝑤}))) |
121 | 78, 82, 104, 111, 120 | en3d 8548 |
. . . . . . . . 9
⊢ (𝜑 → 𝑍 ≈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)) |
122 | | hashen 13710 |
. . . . . . . . . 10
⊢ ((𝑍 ∈ Fin ∧ ((𝑌 / ∼ ) ∩ 𝒫
𝑍) ∈ Fin) →
((♯‘𝑍) =
(♯‘((𝑌 /
∼
) ∩ 𝒫 𝑍))
↔ 𝑍 ≈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍))) |
123 | 73, 35, 122 | syl2anc 586 |
. . . . . . . . 9
⊢ (𝜑 → ((♯‘𝑍) = (♯‘((𝑌 / ∼ ) ∩ 𝒫
𝑍)) ↔ 𝑍 ≈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍))) |
124 | 121, 123 | mpbird 259 |
. . . . . . . 8
⊢ (𝜑 → (♯‘𝑍) = (♯‘((𝑌 / ∼ ) ∩ 𝒫
𝑍))) |
125 | 124 | oveq1d 7173 |
. . . . . . 7
⊢ (𝜑 → ((♯‘𝑍) · 1) =
((♯‘((𝑌
/ ∼ ) ∩ 𝒫
𝑍)) ·
1)) |
126 | 77, 125 | eqtr3d 2860 |
. . . . . 6
⊢ (𝜑 → (♯‘𝑍) = ((♯‘((𝑌 / ∼ ) ∩ 𝒫
𝑍)) ·
1)) |
127 | 38, 70, 126 | 3eqtr4rd 2869 |
. . . . 5
⊢ (𝜑 → (♯‘𝑍) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)(♯‘𝑧)) |
128 | 127 | oveq1d 7173 |
. . . 4
⊢ (𝜑 → ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧)) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫
𝑍)(♯‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧))) |
129 | 31, 32, 128 | 3eqtr4rd 2869 |
. . 3
⊢ (𝜑 → ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧)) = (♯‘𝑌)) |
130 | | hashcl 13720 |
. . . . . 6
⊢ (𝑌 ∈ Fin →
(♯‘𝑌) ∈
ℕ0) |
131 | 5, 130 | syl 17 |
. . . . 5
⊢ (𝜑 → (♯‘𝑌) ∈
ℕ0) |
132 | 131 | nn0cnd 11960 |
. . . 4
⊢ (𝜑 → (♯‘𝑌) ∈
ℂ) |
133 | | diffi 8752 |
. . . . . 6
⊢ ((𝑌 / ∼ ) ∈ Fin →
((𝑌 / ∼ )
∖ 𝒫 𝑍) ∈
Fin) |
134 | 23, 133 | syl 17 |
. . . . 5
⊢ (𝜑 → ((𝑌 / ∼ ) ∖ 𝒫
𝑍) ∈
Fin) |
135 | | eldifi 4105 |
. . . . . 6
⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍) → 𝑧 ∈ (𝑌 / ∼ )) |
136 | 135, 30 | sylan2 594 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)) →
(♯‘𝑧) ∈
ℂ) |
137 | 134, 136 | fsumcl 15092 |
. . . 4
⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧) ∈
ℂ) |
138 | 132, 76, 137 | subaddd 11017 |
. . 3
⊢ (𝜑 → (((♯‘𝑌) − (♯‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧) ↔ ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧)) = (♯‘𝑌))) |
139 | 129, 138 | mpbird 259 |
. 2
⊢ (𝜑 → ((♯‘𝑌) − (♯‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫
𝑍)(♯‘𝑧)) |
140 | 8, 139 | breqtrrd 5096 |
1
⊢ (𝜑 → 𝑃 ∥ ((♯‘𝑌) − (♯‘𝑍))) |