Proof of Theorem stoweidlem16
Step | Hyp | Ref
| Expression |
1 | | stoweidlem16.3 |
. . . 4
⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) |
2 | | simp1 1135 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝜑) |
3 | | fveq1 6775 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑓 → (ℎ‘𝑡) = (𝑓‘𝑡)) |
4 | 3 | breq2d 5088 |
. . . . . . . . . 10
⊢ (ℎ = 𝑓 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑓‘𝑡))) |
5 | 3 | breq1d 5086 |
. . . . . . . . . 10
⊢ (ℎ = 𝑓 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑓‘𝑡) ≤ 1)) |
6 | 4, 5 | anbi12d 631 |
. . . . . . . . 9
⊢ (ℎ = 𝑓 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1))) |
7 | 6 | ralbidv 3119 |
. . . . . . . 8
⊢ (ℎ = 𝑓 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1))) |
8 | | stoweidlem16.2 |
. . . . . . . 8
⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} |
9 | 7, 8 | elrab2 3628 |
. . . . . . 7
⊢ (𝑓 ∈ 𝑌 ↔ (𝑓 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1))) |
10 | 9 | simplbi 498 |
. . . . . 6
⊢ (𝑓 ∈ 𝑌 → 𝑓 ∈ 𝐴) |
11 | 10 | 3ad2ant2 1133 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝑓 ∈ 𝐴) |
12 | | fveq1 6775 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑔 → (ℎ‘𝑡) = (𝑔‘𝑡)) |
13 | 12 | breq2d 5088 |
. . . . . . . . . 10
⊢ (ℎ = 𝑔 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑔‘𝑡))) |
14 | 12 | breq1d 5086 |
. . . . . . . . . 10
⊢ (ℎ = 𝑔 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑔‘𝑡) ≤ 1)) |
15 | 13, 14 | anbi12d 631 |
. . . . . . . . 9
⊢ (ℎ = 𝑔 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))) |
16 | 15 | ralbidv 3119 |
. . . . . . . 8
⊢ (ℎ = 𝑔 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))) |
17 | 16, 8 | elrab2 3628 |
. . . . . . 7
⊢ (𝑔 ∈ 𝑌 ↔ (𝑔 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))) |
18 | 17 | simplbi 498 |
. . . . . 6
⊢ (𝑔 ∈ 𝑌 → 𝑔 ∈ 𝐴) |
19 | 18 | 3ad2ant3 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝑔 ∈ 𝐴) |
20 | | stoweidlem16.5 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
21 | 2, 11, 19, 20 | syl3anc 1370 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
22 | 1, 21 | eqeltrid 2843 |
. . 3
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ 𝐴) |
23 | | stoweidlem16.1 |
. . . . 5
⊢
Ⅎ𝑡𝜑 |
24 | | nfra1 3144 |
. . . . . . . 8
⊢
Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) |
25 | | nfcv 2907 |
. . . . . . . 8
⊢
Ⅎ𝑡𝐴 |
26 | 24, 25 | nfrabw 3317 |
. . . . . . 7
⊢
Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} |
27 | 8, 26 | nfcxfr 2905 |
. . . . . 6
⊢
Ⅎ𝑡𝑌 |
28 | 27 | nfcri 2894 |
. . . . 5
⊢
Ⅎ𝑡 𝑓 ∈ 𝑌 |
29 | 27 | nfcri 2894 |
. . . . 5
⊢
Ⅎ𝑡 𝑔 ∈ 𝑌 |
30 | 23, 28, 29 | nf3an 1904 |
. . . 4
⊢
Ⅎ𝑡(𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) |
31 | 2, 11 | jca 512 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝜑 ∧ 𝑓 ∈ 𝐴)) |
32 | 31 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝜑 ∧ 𝑓 ∈ 𝐴)) |
33 | | stoweidlem16.4 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) |
34 | 32, 33 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 𝑓:𝑇⟶ℝ) |
35 | | simpr 485 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇) |
36 | 34, 35 | ffvelrnd 6964 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑓‘𝑡) ∈ ℝ) |
37 | 2, 19 | jca 512 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝜑 ∧ 𝑔 ∈ 𝐴)) |
38 | | eleq1w 2821 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐴 ↔ 𝑔 ∈ 𝐴)) |
39 | 38 | anbi2d 629 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑔 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝑔 ∈ 𝐴))) |
40 | | feq1 6583 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑔 → (𝑓:𝑇⟶ℝ ↔ 𝑔:𝑇⟶ℝ)) |
41 | 39, 40 | imbi12d 345 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝑔 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝑔 ∈ 𝐴) → 𝑔:𝑇⟶ℝ))) |
42 | 41, 33 | vtoclg 3504 |
. . . . . . . . . 10
⊢ (𝑔 ∈ 𝐴 → ((𝜑 ∧ 𝑔 ∈ 𝐴) → 𝑔:𝑇⟶ℝ)) |
43 | 19, 37, 42 | sylc 65 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝑔:𝑇⟶ℝ) |
44 | 43 | ffvelrnda 6963 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑔‘𝑡) ∈ ℝ) |
45 | 9 | simprbi 497 |
. . . . . . . . . . 11
⊢ (𝑓 ∈ 𝑌 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)) |
46 | 45 | 3ad2ant2 1133 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)) |
47 | 46 | r19.21bi 3134 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)) |
48 | 47 | simpld 495 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑓‘𝑡)) |
49 | 17 | simprbi 497 |
. . . . . . . . . . 11
⊢ (𝑔 ∈ 𝑌 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)) |
50 | 49 | 3ad2ant3 1134 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)) |
51 | 50 | r19.21bi 3134 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)) |
52 | 51 | simpld 495 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑔‘𝑡)) |
53 | 36, 44, 48, 52 | mulge0d 11550 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑓‘𝑡) · (𝑔‘𝑡))) |
54 | 36, 44 | remulcld 11003 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ∈ ℝ) |
55 | 1 | fvmpt2 6888 |
. . . . . . . 8
⊢ ((𝑡 ∈ 𝑇 ∧ ((𝑓‘𝑡) · (𝑔‘𝑡)) ∈ ℝ) → (𝐻‘𝑡) = ((𝑓‘𝑡) · (𝑔‘𝑡))) |
56 | 35, 54, 55 | syl2anc 584 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = ((𝑓‘𝑡) · (𝑔‘𝑡))) |
57 | 53, 56 | breqtrrd 5104 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐻‘𝑡)) |
58 | | 1red 10974 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ) |
59 | 47 | simprd 496 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑓‘𝑡) ≤ 1) |
60 | 51 | simprd 496 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑔‘𝑡) ≤ 1) |
61 | 36, 58, 44, 58, 48, 52, 59, 60 | lemul12ad 11915 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ≤ (1 · 1)) |
62 | | 1t1e1 12133 |
. . . . . . . 8
⊢ (1
· 1) = 1 |
63 | 61, 62 | breqtrdi 5117 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ≤ 1) |
64 | 56, 63 | eqbrtrd 5098 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ≤ 1) |
65 | 57, 64 | jca 512 |
. . . . 5
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)) |
66 | 65 | ex 413 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
67 | 30, 66 | ralrimi 3141 |
. . 3
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)) |
68 | | nfmpt1 5184 |
. . . . . . 7
⊢
Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) |
69 | 1, 68 | nfcxfr 2905 |
. . . . . 6
⊢
Ⅎ𝑡𝐻 |
70 | 69 | nfeq2 2924 |
. . . . 5
⊢
Ⅎ𝑡 ℎ = 𝐻 |
71 | | fveq1 6775 |
. . . . . . 7
⊢ (ℎ = 𝐻 → (ℎ‘𝑡) = (𝐻‘𝑡)) |
72 | 71 | breq2d 5088 |
. . . . . 6
⊢ (ℎ = 𝐻 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝐻‘𝑡))) |
73 | 71 | breq1d 5086 |
. . . . . 6
⊢ (ℎ = 𝐻 → ((ℎ‘𝑡) ≤ 1 ↔ (𝐻‘𝑡) ≤ 1)) |
74 | 72, 73 | anbi12d 631 |
. . . . 5
⊢ (ℎ = 𝐻 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
75 | 70, 74 | ralbid 3160 |
. . . 4
⊢ (ℎ = 𝐻 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
76 | 75 | elrab 3625 |
. . 3
⊢ (𝐻 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ↔ (𝐻 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
77 | 22, 67, 76 | sylanbrc 583 |
. 2
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}) |
78 | 77, 8 | eleqtrrdi 2850 |
1
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ 𝑌) |