Proof of Theorem stoweidlem16
| Step | Hyp | Ref
| Expression |
| 1 | | stoweidlem16.3 |
. . . 4
⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) |
| 2 | | simp1 1137 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝜑) |
| 3 | | fveq1 6905 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑓 → (ℎ‘𝑡) = (𝑓‘𝑡)) |
| 4 | 3 | breq2d 5155 |
. . . . . . . . . 10
⊢ (ℎ = 𝑓 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑓‘𝑡))) |
| 5 | 3 | breq1d 5153 |
. . . . . . . . . 10
⊢ (ℎ = 𝑓 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑓‘𝑡) ≤ 1)) |
| 6 | 4, 5 | anbi12d 632 |
. . . . . . . . 9
⊢ (ℎ = 𝑓 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1))) |
| 7 | 6 | ralbidv 3178 |
. . . . . . . 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 3695 |
. . . . . . 7
⊢ (𝑓 ∈ 𝑌 ↔ (𝑓 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1))) |
| 10 | 9 | simplbi 497 |
. . . . . 6
⊢ (𝑓 ∈ 𝑌 → 𝑓 ∈ 𝐴) |
| 11 | 10 | 3ad2ant2 1135 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝑓 ∈ 𝐴) |
| 12 | | fveq1 6905 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑔 → (ℎ‘𝑡) = (𝑔‘𝑡)) |
| 13 | 12 | breq2d 5155 |
. . . . . . . . . 10
⊢ (ℎ = 𝑔 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑔‘𝑡))) |
| 14 | 12 | breq1d 5153 |
. . . . . . . . . 10
⊢ (ℎ = 𝑔 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑔‘𝑡) ≤ 1)) |
| 15 | 13, 14 | anbi12d 632 |
. . . . . . . . 9
⊢ (ℎ = 𝑔 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))) |
| 16 | 15 | ralbidv 3178 |
. . . . . . . 8
⊢ (ℎ = 𝑔 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))) |
| 17 | 16, 8 | elrab2 3695 |
. . . . . . 7
⊢ (𝑔 ∈ 𝑌 ↔ (𝑔 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1))) |
| 18 | 17 | simplbi 497 |
. . . . . 6
⊢ (𝑔 ∈ 𝑌 → 𝑔 ∈ 𝐴) |
| 19 | 18 | 3ad2ant3 1136 |
. . . . 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 1373 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
| 22 | 1, 21 | eqeltrid 2845 |
. . 3
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ 𝐴) |
| 23 | | stoweidlem16.1 |
. . . . 5
⊢
Ⅎ𝑡𝜑 |
| 24 | | nfra1 3284 |
. . . . . . . 8
⊢
Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) |
| 25 | | nfcv 2905 |
. . . . . . . 8
⊢
Ⅎ𝑡𝐴 |
| 26 | 24, 25 | nfrabw 3475 |
. . . . . . 7
⊢
Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} |
| 27 | 8, 26 | nfcxfr 2903 |
. . . . . 6
⊢
Ⅎ𝑡𝑌 |
| 28 | 27 | nfcri 2897 |
. . . . 5
⊢
Ⅎ𝑡 𝑓 ∈ 𝑌 |
| 29 | 27 | nfcri 2897 |
. . . . 5
⊢
Ⅎ𝑡 𝑔 ∈ 𝑌 |
| 30 | 23, 28, 29 | nf3an 1901 |
. . . 4
⊢
Ⅎ𝑡(𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) |
| 31 | 2, 11 | jca 511 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝜑 ∧ 𝑓 ∈ 𝐴)) |
| 32 | 31 | adantr 480 |
. . . . . . . . . 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 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇) |
| 36 | 34, 35 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑓‘𝑡) ∈ ℝ) |
| 37 | 2, 19 | jca 511 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝜑 ∧ 𝑔 ∈ 𝐴)) |
| 38 | | eleq1w 2824 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐴 ↔ 𝑔 ∈ 𝐴)) |
| 39 | 38 | anbi2d 630 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑔 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝑔 ∈ 𝐴))) |
| 40 | | feq1 6716 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝑔 → (𝑓:𝑇⟶ℝ ↔ 𝑔:𝑇⟶ℝ)) |
| 41 | 39, 40 | imbi12d 344 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝑔 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝑔 ∈ 𝐴) → 𝑔:𝑇⟶ℝ))) |
| 42 | 41, 33 | vtoclg 3554 |
. . . . . . . . . 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 | ffvelcdmda 7104 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑔‘𝑡) ∈ ℝ) |
| 45 | 9 | simprbi 496 |
. . . . . . . . . . 11
⊢ (𝑓 ∈ 𝑌 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)) |
| 46 | 45 | 3ad2ant2 1135 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)) |
| 47 | 46 | r19.21bi 3251 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑓‘𝑡) ∧ (𝑓‘𝑡) ≤ 1)) |
| 48 | 47 | simpld 494 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑓‘𝑡)) |
| 49 | 17 | simprbi 496 |
. . . . . . . . . . 11
⊢ (𝑔 ∈ 𝑌 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)) |
| 50 | 49 | 3ad2ant3 1136 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)) |
| 51 | 50 | r19.21bi 3251 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑔‘𝑡) ∧ (𝑔‘𝑡) ≤ 1)) |
| 52 | 51 | simpld 494 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑔‘𝑡)) |
| 53 | 36, 44, 48, 52 | mulge0d 11840 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑓‘𝑡) · (𝑔‘𝑡))) |
| 54 | 36, 44 | remulcld 11291 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ∈ ℝ) |
| 55 | 1 | fvmpt2 7027 |
. . . . . . . 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 5171 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐻‘𝑡)) |
| 58 | | 1red 11262 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ) |
| 59 | 47 | simprd 495 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑓‘𝑡) ≤ 1) |
| 60 | 51 | simprd 495 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝑔‘𝑡) ≤ 1) |
| 61 | 36, 58, 44, 58, 48, 52, 59, 60 | lemul12ad 12210 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ≤ (1 · 1)) |
| 62 | | 1t1e1 12428 |
. . . . . . . 8
⊢ (1
· 1) = 1 |
| 63 | 61, 62 | breqtrdi 5184 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → ((𝑓‘𝑡) · (𝑔‘𝑡)) ≤ 1) |
| 64 | 56, 63 | eqbrtrd 5165 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) ≤ 1) |
| 65 | 57, 64 | jca 511 |
. . . . 5
⊢ (((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)) |
| 66 | 65 | ex 412 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 → (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
| 67 | 30, 66 | ralrimi 3257 |
. . 3
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1)) |
| 68 | | nfmpt1 5250 |
. . . . . . 7
⊢
Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) |
| 69 | 1, 68 | nfcxfr 2903 |
. . . . . 6
⊢
Ⅎ𝑡𝐻 |
| 70 | 69 | nfeq2 2923 |
. . . . 5
⊢
Ⅎ𝑡 ℎ = 𝐻 |
| 71 | | fveq1 6905 |
. . . . . . 7
⊢ (ℎ = 𝐻 → (ℎ‘𝑡) = (𝐻‘𝑡)) |
| 72 | 71 | breq2d 5155 |
. . . . . 6
⊢ (ℎ = 𝐻 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝐻‘𝑡))) |
| 73 | 71 | breq1d 5153 |
. . . . . 6
⊢ (ℎ = 𝐻 → ((ℎ‘𝑡) ≤ 1 ↔ (𝐻‘𝑡) ≤ 1)) |
| 74 | 72, 73 | anbi12d 632 |
. . . . 5
⊢ (ℎ = 𝐻 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
| 75 | 70, 74 | ralbid 3273 |
. . . 4
⊢ (ℎ = 𝐻 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝐻‘𝑡) ∧ (𝐻‘𝑡) ≤ 1))) |
| 76 | 75 | elrab 3692 |
. . 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 2852 |
1
⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ 𝑌) |