Step | Hyp | Ref
| Expression |
1 | | breq2 3993 |
. . . . . 6
⊢ (𝑗 = 0 → (𝑀 ≤ 𝑗 ↔ 𝑀 ≤ 0)) |
2 | 1 | anbi2d 461 |
. . . . 5
⊢ (𝑗 = 0 → ((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 𝑗) ↔ (𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 0))) |
3 | | fveq2 5496 |
. . . . . 6
⊢ (𝑗 = 0 → (!‘𝑗) =
(!‘0)) |
4 | 3 | breq2d 4001 |
. . . . 5
⊢ (𝑗 = 0 → ((!‘𝑀) ≤ (!‘𝑗) ↔ (!‘𝑀) ≤
(!‘0))) |
5 | 2, 4 | imbi12d 233 |
. . . 4
⊢ (𝑗 = 0 → (((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 𝑗) → (!‘𝑀) ≤ (!‘𝑗)) ↔ ((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 0) → (!‘𝑀) ≤
(!‘0)))) |
6 | | breq2 3993 |
. . . . . 6
⊢ (𝑗 = 𝑘 → (𝑀 ≤ 𝑗 ↔ 𝑀 ≤ 𝑘)) |
7 | 6 | anbi2d 461 |
. . . . 5
⊢ (𝑗 = 𝑘 → ((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 𝑗) ↔ (𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 𝑘))) |
8 | | fveq2 5496 |
. . . . . 6
⊢ (𝑗 = 𝑘 → (!‘𝑗) = (!‘𝑘)) |
9 | 8 | breq2d 4001 |
. . . . 5
⊢ (𝑗 = 𝑘 → ((!‘𝑀) ≤ (!‘𝑗) ↔ (!‘𝑀) ≤ (!‘𝑘))) |
10 | 7, 9 | imbi12d 233 |
. . . 4
⊢ (𝑗 = 𝑘 → (((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 𝑗) → (!‘𝑀) ≤ (!‘𝑗)) ↔ ((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 𝑘) → (!‘𝑀) ≤ (!‘𝑘)))) |
11 | | breq2 3993 |
. . . . . 6
⊢ (𝑗 = (𝑘 + 1) → (𝑀 ≤ 𝑗 ↔ 𝑀 ≤ (𝑘 + 1))) |
12 | 11 | anbi2d 461 |
. . . . 5
⊢ (𝑗 = (𝑘 + 1) → ((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 𝑗) ↔ (𝑀 ∈ ℕ0 ∧ 𝑀 ≤ (𝑘 + 1)))) |
13 | | fveq2 5496 |
. . . . . 6
⊢ (𝑗 = (𝑘 + 1) → (!‘𝑗) = (!‘(𝑘 + 1))) |
14 | 13 | breq2d 4001 |
. . . . 5
⊢ (𝑗 = (𝑘 + 1) → ((!‘𝑀) ≤ (!‘𝑗) ↔ (!‘𝑀) ≤ (!‘(𝑘 + 1)))) |
15 | 12, 14 | imbi12d 233 |
. . . 4
⊢ (𝑗 = (𝑘 + 1) → (((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 𝑗) → (!‘𝑀) ≤ (!‘𝑗)) ↔ ((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ (𝑘 + 1)) → (!‘𝑀) ≤ (!‘(𝑘 + 1))))) |
16 | | breq2 3993 |
. . . . . 6
⊢ (𝑗 = 𝑁 → (𝑀 ≤ 𝑗 ↔ 𝑀 ≤ 𝑁)) |
17 | 16 | anbi2d 461 |
. . . . 5
⊢ (𝑗 = 𝑁 → ((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 𝑗) ↔ (𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁))) |
18 | | fveq2 5496 |
. . . . . 6
⊢ (𝑗 = 𝑁 → (!‘𝑗) = (!‘𝑁)) |
19 | 18 | breq2d 4001 |
. . . . 5
⊢ (𝑗 = 𝑁 → ((!‘𝑀) ≤ (!‘𝑗) ↔ (!‘𝑀) ≤ (!‘𝑁))) |
20 | 17, 19 | imbi12d 233 |
. . . 4
⊢ (𝑗 = 𝑁 → (((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 𝑗) → (!‘𝑀) ≤ (!‘𝑗)) ↔ ((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (!‘𝑀) ≤ (!‘𝑁)))) |
21 | | nn0le0eq0 9163 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ0
→ (𝑀 ≤ 0 ↔
𝑀 = 0)) |
22 | 21 | biimpa 294 |
. . . . . 6
⊢ ((𝑀 ∈ ℕ0
∧ 𝑀 ≤ 0) →
𝑀 = 0) |
23 | 22 | fveq2d 5500 |
. . . . 5
⊢ ((𝑀 ∈ ℕ0
∧ 𝑀 ≤ 0) →
(!‘𝑀) =
(!‘0)) |
24 | | fac0 10662 |
. . . . . . 7
⊢
(!‘0) = 1 |
25 | | 1re 7919 |
. . . . . . 7
⊢ 1 ∈
ℝ |
26 | 24, 25 | eqeltri 2243 |
. . . . . 6
⊢
(!‘0) ∈ ℝ |
27 | 26 | leidi 8404 |
. . . . 5
⊢
(!‘0) ≤ (!‘0) |
28 | 23, 27 | eqbrtrdi 4028 |
. . . 4
⊢ ((𝑀 ∈ ℕ0
∧ 𝑀 ≤ 0) →
(!‘𝑀) ≤
(!‘0)) |
29 | | impexp 261 |
. . . . 5
⊢ (((𝑀 ∈ ℕ0
∧ 𝑀 ≤ 𝑘) → (!‘𝑀) ≤ (!‘𝑘)) ↔ (𝑀 ∈ ℕ0 → (𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘𝑘)))) |
30 | | simpl 108 |
. . . . . . . . . . . . 13
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → 𝑀 ∈
ℕ0) |
31 | 30 | nn0zd 9332 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → 𝑀 ∈ ℤ) |
32 | | peano2nn0 9175 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ (𝑘 + 1) ∈
ℕ0) |
33 | 32 | adantl 275 |
. . . . . . . . . . . . 13
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (𝑘 + 1) ∈
ℕ0) |
34 | 33 | nn0zd 9332 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (𝑘 + 1) ∈ ℤ) |
35 | | zleloe 9259 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) →
(𝑀 ≤ (𝑘 + 1) ↔ (𝑀 < (𝑘 + 1) ∨ 𝑀 = (𝑘 + 1)))) |
36 | 31, 34, 35 | syl2anc 409 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (𝑀 ≤ (𝑘 + 1) ↔ (𝑀 < (𝑘 + 1) ∨ 𝑀 = (𝑘 + 1)))) |
37 | | nn0leltp1 9275 |
. . . . . . . . . . . . 13
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (𝑀 ≤ 𝑘 ↔ 𝑀 < (𝑘 + 1))) |
38 | | faccl 10669 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ℕ0
→ (!‘𝑘) ∈
ℕ) |
39 | 38 | nnred 8891 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℕ0
→ (!‘𝑘) ∈
ℝ) |
40 | | nn0re 9144 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℝ) |
41 | | peano2re 8055 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈
ℝ) |
42 | 40, 41 | syl 14 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℕ0
→ (𝑘 + 1) ∈
ℝ) |
43 | 38 | nnnn0d 9188 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ℕ0
→ (!‘𝑘) ∈
ℕ0) |
44 | 43 | nn0ge0d 9191 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℕ0
→ 0 ≤ (!‘𝑘)) |
45 | | nn0p1nn 9174 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ℕ0
→ (𝑘 + 1) ∈
ℕ) |
46 | 45 | nnge1d 8921 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℕ0
→ 1 ≤ (𝑘 +
1)) |
47 | 39, 42, 44, 46 | lemulge11d 8853 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ0
→ (!‘𝑘) ≤
((!‘𝑘) ·
(𝑘 + 1))) |
48 | | facp1 10664 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ0
→ (!‘(𝑘 + 1)) =
((!‘𝑘) ·
(𝑘 + 1))) |
49 | 47, 48 | breqtrrd 4017 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ0
→ (!‘𝑘) ≤
(!‘(𝑘 +
1))) |
50 | 49 | adantl 275 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (!‘𝑘) ≤ (!‘(𝑘 + 1))) |
51 | | faccl 10669 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑀 ∈ ℕ0
→ (!‘𝑀) ∈
ℕ) |
52 | 51 | nnred 8891 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈ ℕ0
→ (!‘𝑀) ∈
ℝ) |
53 | 52 | adantr 274 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (!‘𝑀) ∈ ℝ) |
54 | 39 | adantl 275 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (!‘𝑘) ∈ ℝ) |
55 | 32 | faccld 10670 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℕ0
→ (!‘(𝑘 + 1))
∈ ℕ) |
56 | 55 | nnred 8891 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ0
→ (!‘(𝑘 + 1))
∈ ℝ) |
57 | 56 | adantl 275 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (!‘(𝑘 + 1)) ∈ ℝ) |
58 | | letr 8002 |
. . . . . . . . . . . . . . . . 17
⊢
(((!‘𝑀) ∈
ℝ ∧ (!‘𝑘)
∈ ℝ ∧ (!‘(𝑘 + 1)) ∈ ℝ) →
(((!‘𝑀) ≤
(!‘𝑘) ∧
(!‘𝑘) ≤
(!‘(𝑘 + 1))) →
(!‘𝑀) ≤
(!‘(𝑘 +
1)))) |
59 | 53, 54, 57, 58 | syl3anc 1233 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (((!‘𝑀) ≤ (!‘𝑘) ∧ (!‘𝑘) ≤ (!‘(𝑘 + 1))) → (!‘𝑀) ≤ (!‘(𝑘 + 1)))) |
60 | 50, 59 | mpan2d 426 |
. . . . . . . . . . . . . . 15
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → ((!‘𝑀) ≤ (!‘𝑘) → (!‘𝑀) ≤ (!‘(𝑘 + 1)))) |
61 | 60 | imim2d 54 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → ((𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘𝑘)) → (𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘(𝑘 + 1))))) |
62 | 61 | com23 78 |
. . . . . . . . . . . . 13
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (𝑀 ≤ 𝑘 → ((𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘𝑘)) → (!‘𝑀) ≤ (!‘(𝑘 + 1))))) |
63 | 37, 62 | sylbird 169 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (𝑀 < (𝑘 + 1) → ((𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘𝑘)) → (!‘𝑀) ≤ (!‘(𝑘 + 1))))) |
64 | | fveq2 5496 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 = (𝑘 + 1) → (!‘𝑀) = (!‘(𝑘 + 1))) |
65 | 52 | leidd 8433 |
. . . . . . . . . . . . . . . 16
⊢ (𝑀 ∈ ℕ0
→ (!‘𝑀) ≤
(!‘𝑀)) |
66 | | breq2 3993 |
. . . . . . . . . . . . . . . 16
⊢
((!‘𝑀) =
(!‘(𝑘 + 1)) →
((!‘𝑀) ≤
(!‘𝑀) ↔
(!‘𝑀) ≤
(!‘(𝑘 +
1)))) |
67 | 65, 66 | syl5ibcom 154 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈ ℕ0
→ ((!‘𝑀) =
(!‘(𝑘 + 1)) →
(!‘𝑀) ≤
(!‘(𝑘 +
1)))) |
68 | 64, 67 | syl5 32 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ ℕ0
→ (𝑀 = (𝑘 + 1) → (!‘𝑀) ≤ (!‘(𝑘 + 1)))) |
69 | 68 | adantr 274 |
. . . . . . . . . . . . 13
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (𝑀 = (𝑘 + 1) → (!‘𝑀) ≤ (!‘(𝑘 + 1)))) |
70 | 69 | a1dd 48 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (𝑀 = (𝑘 + 1) → ((𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘𝑘)) → (!‘𝑀) ≤ (!‘(𝑘 + 1))))) |
71 | 63, 70 | jaod 712 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → ((𝑀 < (𝑘 + 1) ∨ 𝑀 = (𝑘 + 1)) → ((𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘𝑘)) → (!‘𝑀) ≤ (!‘(𝑘 + 1))))) |
72 | 36, 71 | sylbid 149 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℕ0
∧ 𝑘 ∈
ℕ0) → (𝑀 ≤ (𝑘 + 1) → ((𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘𝑘)) → (!‘𝑀) ≤ (!‘(𝑘 + 1))))) |
73 | 72 | ex 114 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℕ0
→ (𝑘 ∈
ℕ0 → (𝑀 ≤ (𝑘 + 1) → ((𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘𝑘)) → (!‘𝑀) ≤ (!‘(𝑘 + 1)))))) |
74 | 73 | com13 80 |
. . . . . . . 8
⊢ (𝑀 ≤ (𝑘 + 1) → (𝑘 ∈ ℕ0 → (𝑀 ∈ ℕ0
→ ((𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘𝑘)) → (!‘𝑀) ≤ (!‘(𝑘 + 1)))))) |
75 | 74 | com4l 84 |
. . . . . . 7
⊢ (𝑘 ∈ ℕ0
→ (𝑀 ∈
ℕ0 → ((𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘𝑘)) → (𝑀 ≤ (𝑘 + 1) → (!‘𝑀) ≤ (!‘(𝑘 + 1)))))) |
76 | 75 | a2d 26 |
. . . . . 6
⊢ (𝑘 ∈ ℕ0
→ ((𝑀 ∈
ℕ0 → (𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘𝑘))) → (𝑀 ∈ ℕ0 → (𝑀 ≤ (𝑘 + 1) → (!‘𝑀) ≤ (!‘(𝑘 + 1)))))) |
77 | 76 | imp4a 347 |
. . . . 5
⊢ (𝑘 ∈ ℕ0
→ ((𝑀 ∈
ℕ0 → (𝑀 ≤ 𝑘 → (!‘𝑀) ≤ (!‘𝑘))) → ((𝑀 ∈ ℕ0 ∧ 𝑀 ≤ (𝑘 + 1)) → (!‘𝑀) ≤ (!‘(𝑘 + 1))))) |
78 | 29, 77 | syl5bi 151 |
. . . 4
⊢ (𝑘 ∈ ℕ0
→ (((𝑀 ∈
ℕ0 ∧ 𝑀
≤ 𝑘) →
(!‘𝑀) ≤
(!‘𝑘)) → ((𝑀 ∈ ℕ0
∧ 𝑀 ≤ (𝑘 + 1)) → (!‘𝑀) ≤ (!‘(𝑘 + 1))))) |
79 | 5, 10, 15, 20, 28, 78 | nn0ind 9326 |
. . 3
⊢ (𝑁 ∈ ℕ0
→ ((𝑀 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) →
(!‘𝑀) ≤
(!‘𝑁))) |
80 | 79 | 3impib 1196 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) →
(!‘𝑀) ≤
(!‘𝑁)) |
81 | 80 | 3com12 1202 |
1
⊢ ((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0 ∧ 𝑀
≤ 𝑁) →
(!‘𝑀) ≤
(!‘𝑁)) |