Proof of Theorem stoweidlem22
| Step | Hyp | Ref
| Expression |
| 1 | | stoweidlem22.8 |
. . . 4
⊢
Ⅎ𝑡𝜑 |
| 2 | | stoweidlem22.9 |
. . . . 5
⊢
Ⅎ𝑡𝐹 |
| 3 | 2 | nfel1 2922 |
. . . 4
⊢
Ⅎ𝑡 𝐹 ∈ 𝐴 |
| 4 | | stoweidlem22.10 |
. . . . 5
⊢
Ⅎ𝑡𝐺 |
| 5 | 4 | nfel1 2922 |
. . . 4
⊢
Ⅎ𝑡 𝐺 ∈ 𝐴 |
| 6 | 1, 3, 5 | nf3an 1901 |
. . 3
⊢
Ⅎ𝑡(𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) |
| 7 | | simpr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇) |
| 8 | | simpl1 1192 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝜑) |
| 9 | | stoweidlem22.2 |
. . . . . . . . . . 11
⊢ 𝐼 = (𝑡 ∈ 𝑇 ↦ -1) |
| 10 | | neg1rr 12381 |
. . . . . . . . . . . 12
⊢ -1 ∈
ℝ |
| 11 | | stoweidlem22.7 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) |
| 12 | 11 | stoweidlem4 46019 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ -1 ∈ ℝ) →
(𝑡 ∈ 𝑇 ↦ -1) ∈ 𝐴) |
| 13 | 10, 12 | mpan2 691 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ -1) ∈ 𝐴) |
| 14 | 9, 13 | eqeltrid 2845 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 ∈ 𝐴) |
| 15 | | eleq1 2829 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝐼 → (𝑓 ∈ 𝐴 ↔ 𝐼 ∈ 𝐴)) |
| 16 | 15 | anbi2d 630 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝐼 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐼 ∈ 𝐴))) |
| 17 | | feq1 6716 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝐼 → (𝑓:𝑇⟶ℝ ↔ 𝐼:𝑇⟶ℝ)) |
| 18 | 16, 17 | imbi12d 344 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝐼 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐼 ∈ 𝐴) → 𝐼:𝑇⟶ℝ))) |
| 19 | | stoweidlem22.4 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) |
| 20 | 18, 19 | vtoclg 3554 |
. . . . . . . . . . 11
⊢ (𝐼 ∈ 𝐴 → ((𝜑 ∧ 𝐼 ∈ 𝐴) → 𝐼:𝑇⟶ℝ)) |
| 21 | 20 | anabsi7 671 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐼 ∈ 𝐴) → 𝐼:𝑇⟶ℝ) |
| 22 | 8, 14, 21 | syl2anc2 585 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐼:𝑇⟶ℝ) |
| 23 | 22, 7 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐼‘𝑡) ∈ ℝ) |
| 24 | | simpl3 1194 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐺 ∈ 𝐴) |
| 25 | | eleq1 2829 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = 𝐺 → (𝑓 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴)) |
| 26 | 25 | anbi2d 630 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝐺 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐺 ∈ 𝐴))) |
| 27 | | feq1 6716 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝐺 → (𝑓:𝑇⟶ℝ ↔ 𝐺:𝑇⟶ℝ)) |
| 28 | 26, 27 | imbi12d 344 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝐺 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ))) |
| 29 | 28, 19 | vtoclg 3554 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ)) |
| 30 | 29 | anabsi7 671 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ) |
| 31 | 30 | 3adant3 1133 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → 𝐺:𝑇⟶ℝ) |
| 32 | | simp3 1139 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇) |
| 33 | 31, 32 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ) |
| 34 | 8, 24, 7, 33 | syl3anc 1373 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ) |
| 35 | 23, 34 | remulcld 11291 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐼‘𝑡) · (𝐺‘𝑡)) ∈ ℝ) |
| 36 | | stoweidlem22.3 |
. . . . . . . 8
⊢ 𝐿 = (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) |
| 37 | 36 | fvmpt2 7027 |
. . . . . . 7
⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐼‘𝑡) · (𝐺‘𝑡)) ∈ ℝ) → (𝐿‘𝑡) = ((𝐼‘𝑡) · (𝐺‘𝑡))) |
| 38 | 7, 35, 37 | syl2anc 584 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐿‘𝑡) = ((𝐼‘𝑡) · (𝐺‘𝑡))) |
| 39 | 9 | fvmpt2 7027 |
. . . . . . . . 9
⊢ ((𝑡 ∈ 𝑇 ∧ -1 ∈ ℝ) → (𝐼‘𝑡) = -1) |
| 40 | 10, 39 | mpan2 691 |
. . . . . . . 8
⊢ (𝑡 ∈ 𝑇 → (𝐼‘𝑡) = -1) |
| 41 | 40 | adantl 481 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐼‘𝑡) = -1) |
| 42 | 41 | oveq1d 7446 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐼‘𝑡) · (𝐺‘𝑡)) = (-1 · (𝐺‘𝑡))) |
| 43 | 34 | recnd 11289 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℂ) |
| 44 | 43 | mulm1d 11715 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (-1 · (𝐺‘𝑡)) = -(𝐺‘𝑡)) |
| 45 | 38, 42, 44 | 3eqtrd 2781 |
. . . . 5
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐿‘𝑡) = -(𝐺‘𝑡)) |
| 46 | 45 | oveq2d 7447 |
. . . 4
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + (𝐿‘𝑡)) = ((𝐹‘𝑡) + -(𝐺‘𝑡))) |
| 47 | | simpl2 1193 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐹 ∈ 𝐴) |
| 48 | | eleq1 2829 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝐹 → (𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴)) |
| 49 | 48 | anbi2d 630 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝐹 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴))) |
| 50 | | feq1 6716 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝐹 → (𝑓:𝑇⟶ℝ ↔ 𝐹:𝑇⟶ℝ)) |
| 51 | 49, 50 | imbi12d 344 |
. . . . . . . . . 10
⊢ (𝑓 = 𝐹 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ))) |
| 52 | 51, 19 | vtoclg 3554 |
. . . . . . . . 9
⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ)) |
| 53 | 52 | anabsi7 671 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ) |
| 54 | 8, 47, 53 | syl2anc 584 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐹:𝑇⟶ℝ) |
| 55 | 54, 7 | ffvelcdmd 7105 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ) |
| 56 | 55 | recnd 11289 |
. . . . 5
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ) |
| 57 | 56, 43 | negsubd 11626 |
. . . 4
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + -(𝐺‘𝑡)) = ((𝐹‘𝑡) − (𝐺‘𝑡))) |
| 58 | 46, 57 | eqtr2d 2778 |
. . 3
⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) − (𝐺‘𝑡)) = ((𝐹‘𝑡) + (𝐿‘𝑡))) |
| 59 | 6, 58 | mpteq2da 5240 |
. 2
⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡)))) |
| 60 | 14 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → 𝐼 ∈ 𝐴) |
| 61 | | nfmpt1 5250 |
. . . . . . . 8
⊢
Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ -1) |
| 62 | 9, 61 | nfcxfr 2903 |
. . . . . . 7
⊢
Ⅎ𝑡𝐼 |
| 63 | 62 | nfeq2 2923 |
. . . . . 6
⊢
Ⅎ𝑡 𝑓 = 𝐼 |
| 64 | 4 | nfeq2 2923 |
. . . . . 6
⊢
Ⅎ𝑡 𝑔 = 𝐺 |
| 65 | | stoweidlem22.6 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
| 66 | 63, 64, 65 | stoweidlem6 46021 |
. . . . 5
⊢ ((𝜑 ∧ 𝐼 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴) |
| 67 | 60, 66 | syld3an2 1413 |
. . . 4
⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴) |
| 68 | 36, 67 | eqeltrid 2845 |
. . 3
⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → 𝐿 ∈ 𝐴) |
| 69 | | stoweidlem22.5 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
| 70 | | nfmpt1 5250 |
. . . . 5
⊢
Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) |
| 71 | 36, 70 | nfcxfr 2903 |
. . . 4
⊢
Ⅎ𝑡𝐿 |
| 72 | 69, 2, 71 | stoweidlem8 46023 |
. . 3
⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐿 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))) ∈ 𝐴) |
| 73 | 68, 72 | syld3an3 1411 |
. 2
⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))) ∈ 𝐴) |
| 74 | 59, 73 | eqeltrd 2841 |
1
⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) ∈ 𝐴) |