| Step | Hyp | Ref
| Expression |
| 1 | | gsumzadd.g |
. . . . . 6
⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 2 | | gsumzadd.b |
. . . . . . . 8
⊢ 𝐵 = (Base‘𝐺) |
| 3 | | gsumzadd.0 |
. . . . . . . 8
⊢ 0 =
(0g‘𝐺) |
| 4 | 2, 3 | mndidcl 18762 |
. . . . . . 7
⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
| 5 | 1, 4 | syl 17 |
. . . . . 6
⊢ (𝜑 → 0 ∈ 𝐵) |
| 6 | | gsumzadd.p |
. . . . . . 7
⊢ + =
(+g‘𝐺) |
| 7 | 2, 6, 3 | mndlid 18767 |
. . . . . 6
⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 ) |
| 8 | 1, 5, 7 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ( 0 + 0 ) = 0 ) |
| 9 | 8 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑊 = ∅) → ( 0 + 0 ) = 0 ) |
| 10 | | gsumzaddlem.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
| 11 | | gsumzadd.a |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| 12 | 3 | fvexi 6920 |
. . . . . . . . 9
⊢ 0 ∈
V |
| 13 | 12 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈ V) |
| 14 | | gsumzaddlem.h |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐻:𝐴⟶𝐵) |
| 15 | 14, 11 | fexd 7247 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐻 ∈ V) |
| 16 | 15 | suppun 8209 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 )) |
| 17 | | gsumzaddlem.w |
. . . . . . . . 9
⊢ 𝑊 = ((𝐹 ∪ 𝐻) supp 0 ) |
| 18 | 16, 17 | sseqtrrdi 4025 |
. . . . . . . 8
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊) |
| 19 | 10, 11, 13, 18 | gsumcllem 19926 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 = (𝑥 ∈ 𝐴 ↦ 0 )) |
| 20 | 19 | oveq2d 7447 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 ))) |
| 21 | 3 | gsumz 18849 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 ) |
| 22 | 1, 11, 21 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 ) |
| 23 | 22 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 )) = 0 ) |
| 24 | 20, 23 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg 𝐹) = 0 ) |
| 25 | 10, 11 | fexd 7247 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 ∈ V) |
| 26 | 25 | suppun 8209 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐻 supp 0 ) ⊆ ((𝐻 ∪ 𝐹) supp 0 )) |
| 27 | | uncom 4158 |
. . . . . . . . . . 11
⊢ (𝐹 ∪ 𝐻) = (𝐻 ∪ 𝐹) |
| 28 | 27 | oveq1i 7441 |
. . . . . . . . . 10
⊢ ((𝐹 ∪ 𝐻) supp 0 ) = ((𝐻 ∪ 𝐹) supp 0 ) |
| 29 | 26, 28 | sseqtrrdi 4025 |
. . . . . . . . 9
⊢ (𝜑 → (𝐻 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 )) |
| 30 | 29, 17 | sseqtrrdi 4025 |
. . . . . . . 8
⊢ (𝜑 → (𝐻 supp 0 ) ⊆ 𝑊) |
| 31 | 14, 11, 13, 30 | gsumcllem 19926 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐻 = (𝑥 ∈ 𝐴 ↦ 0 )) |
| 32 | 31 | oveq2d 7447 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg 𝐻) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 ))) |
| 33 | 32, 23 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg 𝐻) = 0 ) |
| 34 | 24, 33 | oveq12d 7449 |
. . . 4
⊢ ((𝜑 ∧ 𝑊 = ∅) → ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)) = ( 0 + 0 )) |
| 35 | 11 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐴 ∈ 𝑉) |
| 36 | 5 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ 𝐴) → 0 ∈ 𝐵) |
| 37 | 35, 36, 36, 19, 31 | offval2 7717 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐹 ∘f + 𝐻) = (𝑥 ∈ 𝐴 ↦ ( 0 + 0 ))) |
| 38 | 9 | mpteq2dv 5244 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑥 ∈ 𝐴 ↦ ( 0 + 0 )) = (𝑥 ∈ 𝐴 ↦ 0 )) |
| 39 | 37, 38 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐹 ∘f + 𝐻) = (𝑥 ∈ 𝐴 ↦ 0 )) |
| 40 | 39 | oveq2d 7447 |
. . . . 5
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg (𝐹 ∘f + 𝐻)) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 0 ))) |
| 41 | 40, 23 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg (𝐹 ∘f + 𝐻)) = 0 ) |
| 42 | 9, 34, 41 | 3eqtr4rd 2788 |
. . 3
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
| 43 | 42 | ex 412 |
. 2
⊢ (𝜑 → (𝑊 = ∅ → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))) |
| 44 | 1 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐺 ∈ Mnd) |
| 45 | 2, 6 | mndcl 18755 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ Mnd ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → (𝑧 + 𝑤) ∈ 𝐵) |
| 46 | 45 | 3expb 1121 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Mnd ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧 + 𝑤) ∈ 𝐵) |
| 47 | 44, 46 | sylan 580 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧 + 𝑤) ∈ 𝐵) |
| 48 | 47 | caovclg 7625 |
. . . . . . 7
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵) |
| 49 | | simprl 771 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (♯‘𝑊) ∈
ℕ) |
| 50 | | nnuz 12921 |
. . . . . . . 8
⊢ ℕ =
(ℤ≥‘1) |
| 51 | 49, 50 | eleqtrdi 2851 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (♯‘𝑊) ∈
(ℤ≥‘1)) |
| 52 | 10 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐹:𝐴⟶𝐵) |
| 53 | | f1of1 6847 |
. . . . . . . . . . . 12
⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → 𝑓:(1...(♯‘𝑊))–1-1→𝑊) |
| 54 | 53 | ad2antll 729 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1→𝑊) |
| 55 | | suppssdm 8202 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∪ 𝐻) supp 0 ) ⊆ dom (𝐹 ∪ 𝐻) |
| 56 | 55 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐹 ∪ 𝐻) supp 0 ) ⊆ dom (𝐹 ∪ 𝐻)) |
| 57 | 17 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑊 = ((𝐹 ∪ 𝐻) supp 0 )) |
| 58 | | dmun 5921 |
. . . . . . . . . . . . . 14
⊢ dom
(𝐹 ∪ 𝐻) = (dom 𝐹 ∪ dom 𝐻) |
| 59 | 10 | fdmd 6746 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → dom 𝐹 = 𝐴) |
| 60 | 14 | fdmd 6746 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → dom 𝐻 = 𝐴) |
| 61 | 59, 60 | uneq12d 4169 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (dom 𝐹 ∪ dom 𝐻) = (𝐴 ∪ 𝐴)) |
| 62 | | unidm 4157 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∪ 𝐴) = 𝐴 |
| 63 | 61, 62 | eqtrdi 2793 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (dom 𝐹 ∪ dom 𝐻) = 𝐴) |
| 64 | 58, 63 | eqtr2id 2790 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐴 = dom (𝐹 ∪ 𝐻)) |
| 65 | 56, 57, 64 | 3sstr4d 4039 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑊 ⊆ 𝐴) |
| 66 | 65 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑊 ⊆ 𝐴) |
| 67 | | f1ss 6809 |
. . . . . . . . . . 11
⊢ ((𝑓:(1...(♯‘𝑊))–1-1→𝑊 ∧ 𝑊 ⊆ 𝐴) → 𝑓:(1...(♯‘𝑊))–1-1→𝐴) |
| 68 | 54, 66, 67 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1→𝐴) |
| 69 | | f1f 6804 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘𝑊))–1-1→𝐴 → 𝑓:(1...(♯‘𝑊))⟶𝐴) |
| 70 | 68, 69 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑓:(1...(♯‘𝑊))⟶𝐴) |
| 71 | | fco 6760 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(♯‘𝑊))⟶𝐵) |
| 72 | 52, 70, 71 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐹 ∘ 𝑓):(1...(♯‘𝑊))⟶𝐵) |
| 73 | 72 | ffvelcdmda 7104 |
. . . . . . 7
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) ∈ 𝐵) |
| 74 | 14 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐻:𝐴⟶𝐵) |
| 75 | | fco 6760 |
. . . . . . . . 9
⊢ ((𝐻:𝐴⟶𝐵 ∧ 𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐻 ∘ 𝑓):(1...(♯‘𝑊))⟶𝐵) |
| 76 | 74, 70, 75 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐻 ∘ 𝑓):(1...(♯‘𝑊))⟶𝐵) |
| 77 | 76 | ffvelcdmda 7104 |
. . . . . . 7
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐻 ∘ 𝑓)‘𝑘) ∈ 𝐵) |
| 78 | 52 | ffnd 6737 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐹 Fn 𝐴) |
| 79 | 74 | ffnd 6737 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐻 Fn 𝐴) |
| 80 | 11 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐴 ∈ 𝑉) |
| 81 | | ovexd 7466 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) →
(1...(♯‘𝑊))
∈ V) |
| 82 | | inidm 4227 |
. . . . . . . . . . 11
⊢ (𝐴 ∩ 𝐴) = 𝐴 |
| 83 | 78, 79, 70, 80, 80, 81, 82 | ofco 7722 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ((𝐹 ∘f + 𝐻) ∘ 𝑓) = ((𝐹 ∘ 𝑓) ∘f + (𝐻 ∘ 𝑓))) |
| 84 | 83 | fveq1d 6908 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (((𝐹 ∘f + 𝐻) ∘ 𝑓)‘𝑘) = (((𝐹 ∘ 𝑓) ∘f + (𝐻 ∘ 𝑓))‘𝑘)) |
| 85 | 84 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (((𝐹 ∘f + 𝐻) ∘ 𝑓)‘𝑘) = (((𝐹 ∘ 𝑓) ∘f + (𝐻 ∘ 𝑓))‘𝑘)) |
| 86 | | fnfco 6773 |
. . . . . . . . . 10
⊢ ((𝐹 Fn 𝐴 ∧ 𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐹 ∘ 𝑓) Fn (1...(♯‘𝑊))) |
| 87 | 78, 70, 86 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐹 ∘ 𝑓) Fn (1...(♯‘𝑊))) |
| 88 | | fnfco 6773 |
. . . . . . . . . 10
⊢ ((𝐻 Fn 𝐴 ∧ 𝑓:(1...(♯‘𝑊))⟶𝐴) → (𝐻 ∘ 𝑓) Fn (1...(♯‘𝑊))) |
| 89 | 79, 70, 88 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐻 ∘ 𝑓) Fn (1...(♯‘𝑊))) |
| 90 | | inidm 4227 |
. . . . . . . . 9
⊢
((1...(♯‘𝑊)) ∩ (1...(♯‘𝑊))) = (1...(♯‘𝑊)) |
| 91 | | eqidd 2738 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) = ((𝐹 ∘ 𝑓)‘𝑘)) |
| 92 | | eqidd 2738 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐻 ∘ 𝑓)‘𝑘) = ((𝐻 ∘ 𝑓)‘𝑘)) |
| 93 | 87, 89, 81, 81, 90, 91, 92 | ofval 7708 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (((𝐹 ∘ 𝑓) ∘f + (𝐻 ∘ 𝑓))‘𝑘) = (((𝐹 ∘ 𝑓)‘𝑘) + ((𝐻 ∘ 𝑓)‘𝑘))) |
| 94 | 85, 93 | eqtrd 2777 |
. . . . . . 7
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (((𝐹 ∘f + 𝐻) ∘ 𝑓)‘𝑘) = (((𝐹 ∘ 𝑓)‘𝑘) + ((𝐻 ∘ 𝑓)‘𝑘))) |
| 95 | 1 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝐺 ∈ Mnd) |
| 96 | | elfzouz 13703 |
. . . . . . . . . 10
⊢ (𝑛 ∈
(1..^(♯‘𝑊))
→ 𝑛 ∈
(ℤ≥‘1)) |
| 97 | 96 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑛 ∈
(ℤ≥‘1)) |
| 98 | | elfzouz2 13714 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈
(1..^(♯‘𝑊))
→ (♯‘𝑊)
∈ (ℤ≥‘𝑛)) |
| 99 | 98 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (♯‘𝑊) ∈ (ℤ≥‘𝑛)) |
| 100 | | fzss2 13604 |
. . . . . . . . . . . 12
⊢
((♯‘𝑊)
∈ (ℤ≥‘𝑛) → (1...𝑛) ⊆ (1...(♯‘𝑊))) |
| 101 | 99, 100 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (1...𝑛) ⊆ (1...(♯‘𝑊))) |
| 102 | 101 | sselda 3983 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ (1...(♯‘𝑊))) |
| 103 | 73 | adantlr 715 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) ∈ 𝐵) |
| 104 | 102, 103 | syldan 591 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → ((𝐹 ∘ 𝑓)‘𝑘) ∈ 𝐵) |
| 105 | 2, 6 | mndcl 18755 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Mnd ∧ 𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑘 + 𝑥) ∈ 𝐵) |
| 106 | 105 | 3expb 1121 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ Mnd ∧ (𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑘 + 𝑥) ∈ 𝐵) |
| 107 | 95, 106 | sylan 580 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ (𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑘 + 𝑥) ∈ 𝐵) |
| 108 | 97, 104, 107 | seqcl 14063 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐹 ∘ 𝑓))‘𝑛) ∈ 𝐵) |
| 109 | 77 | adantlr 715 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐻 ∘ 𝑓)‘𝑘) ∈ 𝐵) |
| 110 | 102, 109 | syldan 591 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → ((𝐻 ∘ 𝑓)‘𝑘) ∈ 𝐵) |
| 111 | 97, 110, 107 | seqcl 14063 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐻 ∘ 𝑓))‘𝑛) ∈ 𝐵) |
| 112 | | fzofzp1 13803 |
. . . . . . . . 9
⊢ (𝑛 ∈
(1..^(♯‘𝑊))
→ (𝑛 + 1) ∈
(1...(♯‘𝑊))) |
| 113 | | ffvelcdm 7101 |
. . . . . . . . 9
⊢ (((𝐹 ∘ 𝑓):(1...(♯‘𝑊))⟶𝐵 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘(𝑛 + 1)) ∈ 𝐵) |
| 114 | 72, 112, 113 | syl2an 596 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘(𝑛 + 1)) ∈ 𝐵) |
| 115 | | ffvelcdm 7101 |
. . . . . . . . 9
⊢ (((𝐻 ∘ 𝑓):(1...(♯‘𝑊))⟶𝐵 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → ((𝐻 ∘ 𝑓)‘(𝑛 + 1)) ∈ 𝐵) |
| 116 | 76, 112, 115 | syl2an 596 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐻 ∘ 𝑓)‘(𝑛 + 1)) ∈ 𝐵) |
| 117 | | fvco3 7008 |
. . . . . . . . . . . 12
⊢ ((𝑓:(1...(♯‘𝑊))⟶𝐴 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘(𝑛 + 1)) = (𝐹‘(𝑓‘(𝑛 + 1)))) |
| 118 | 70, 112, 117 | syl2an 596 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘(𝑛 + 1)) = (𝐹‘(𝑓‘(𝑛 + 1)))) |
| 119 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (𝑓‘(𝑛 + 1)) → (𝐹‘𝑘) = (𝐹‘(𝑓‘(𝑛 + 1)))) |
| 120 | 119 | eleq1d 2826 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑓‘(𝑛 + 1)) → ((𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) ↔ (𝐹‘(𝑓‘(𝑛 + 1))) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))) |
| 121 | | gsumzaddlem.4 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑘 ∈ (𝐴 ∖ 𝑥))) → (𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))})) |
| 122 | 121 | expr 456 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (𝑘 ∈ (𝐴 ∖ 𝑥) → (𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}))) |
| 123 | 122 | ralrimiv 3145 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → ∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))})) |
| 124 | 123 | ex 412 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑥 ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}))) |
| 125 | 124 | alrimiv 1927 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑥(𝑥 ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}))) |
| 126 | 125 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ∀𝑥(𝑥 ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}))) |
| 127 | | imassrn 6089 |
. . . . . . . . . . . . . 14
⊢ (𝑓 “ (1...𝑛)) ⊆ ran 𝑓 |
| 128 | 70 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑓:(1...(♯‘𝑊))⟶𝐴) |
| 129 | 128 | frnd 6744 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ran 𝑓 ⊆ 𝐴) |
| 130 | 127, 129 | sstrid 3995 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 “ (1...𝑛)) ⊆ 𝐴) |
| 131 | | vex 3484 |
. . . . . . . . . . . . . . 15
⊢ 𝑓 ∈ V |
| 132 | 131 | imaex 7936 |
. . . . . . . . . . . . . 14
⊢ (𝑓 “ (1...𝑛)) ∈ V |
| 133 | | sseq1 4009 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑓 “ (1...𝑛)) → (𝑥 ⊆ 𝐴 ↔ (𝑓 “ (1...𝑛)) ⊆ 𝐴)) |
| 134 | | difeq2 4120 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑓 “ (1...𝑛)) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝑓 “ (1...𝑛)))) |
| 135 | | reseq2 5992 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = (𝑓 “ (1...𝑛)) → (𝐻 ↾ 𝑥) = (𝐻 ↾ (𝑓 “ (1...𝑛)))) |
| 136 | 135 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = (𝑓 “ (1...𝑛)) → (𝐺 Σg (𝐻 ↾ 𝑥)) = (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))) |
| 137 | 136 | sneqd 4638 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = (𝑓 “ (1...𝑛)) → {(𝐺 Σg (𝐻 ↾ 𝑥))} = {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) |
| 138 | 137 | fveq2d 6910 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = (𝑓 “ (1...𝑛)) → (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}) = (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})) |
| 139 | 138 | eleq2d 2827 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑓 “ (1...𝑛)) → ((𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}) ↔ (𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))) |
| 140 | 134, 139 | raleqbidv 3346 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑓 “ (1...𝑛)) → (∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))}) ↔ ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))) |
| 141 | 133, 140 | imbi12d 344 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝑓 “ (1...𝑛)) → ((𝑥 ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))})) ↔ ((𝑓 “ (1...𝑛)) ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})))) |
| 142 | 132, 141 | spcv 3605 |
. . . . . . . . . . . . 13
⊢
(∀𝑥(𝑥 ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ 𝑥)(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ 𝑥))})) → ((𝑓 “ (1...𝑛)) ⊆ 𝐴 → ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}))) |
| 143 | 126, 130,
142 | sylc 65 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ∀𝑘 ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))(𝐹‘𝑘) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})) |
| 144 | | ffvelcdm 7101 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:(1...(♯‘𝑊))⟶𝐴 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊))) → (𝑓‘(𝑛 + 1)) ∈ 𝐴) |
| 145 | 70, 112, 144 | syl2an 596 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓‘(𝑛 + 1)) ∈ 𝐴) |
| 146 | | fzp1nel 13651 |
. . . . . . . . . . . . . 14
⊢ ¬
(𝑛 + 1) ∈ (1...𝑛) |
| 147 | 68 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑓:(1...(♯‘𝑊))–1-1→𝐴) |
| 148 | 112 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑛 + 1) ∈ (1...(♯‘𝑊))) |
| 149 | | f1elima 7283 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:(1...(♯‘𝑊))–1-1→𝐴 ∧ (𝑛 + 1) ∈ (1...(♯‘𝑊)) ∧ (1...𝑛) ⊆ (1...(♯‘𝑊))) → ((𝑓‘(𝑛 + 1)) ∈ (𝑓 “ (1...𝑛)) ↔ (𝑛 + 1) ∈ (1...𝑛))) |
| 150 | 147, 148,
101, 149 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝑓‘(𝑛 + 1)) ∈ (𝑓 “ (1...𝑛)) ↔ (𝑛 + 1) ∈ (1...𝑛))) |
| 151 | 146, 150 | mtbiri 327 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ¬ (𝑓‘(𝑛 + 1)) ∈ (𝑓 “ (1...𝑛))) |
| 152 | 145, 151 | eldifd 3962 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓‘(𝑛 + 1)) ∈ (𝐴 ∖ (𝑓 “ (1...𝑛)))) |
| 153 | 120, 143,
152 | rspcdva 3623 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝐹‘(𝑓‘(𝑛 + 1))) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})) |
| 154 | 118, 153 | eqeltrd 2841 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘(𝑛 + 1)) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))})) |
| 155 | | gsumzadd.z |
. . . . . . . . . . . . 13
⊢ 𝑍 = (Cntz‘𝐺) |
| 156 | 132 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 “ (1...𝑛)) ∈ V) |
| 157 | 14 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝐻:𝐴⟶𝐵) |
| 158 | 157, 130 | fssresd 6775 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝐻 ↾ (𝑓 “ (1...𝑛))):(𝑓 “ (1...𝑛))⟶𝐵) |
| 159 | | gsumzaddlem.2 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ran 𝐻 ⊆ (𝑍‘ran 𝐻)) |
| 160 | 159 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ran 𝐻 ⊆ (𝑍‘ran 𝐻)) |
| 161 | | resss 6019 |
. . . . . . . . . . . . . . 15
⊢ (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ 𝐻 |
| 162 | 161 | rnssi 5951 |
. . . . . . . . . . . . . 14
⊢ ran
(𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ ran 𝐻 |
| 163 | 155 | cntzidss 19358 |
. . . . . . . . . . . . . 14
⊢ ((ran
𝐻 ⊆ (𝑍‘ran 𝐻) ∧ ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ ran 𝐻) → ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ (𝑍‘ran (𝐻 ↾ (𝑓 “ (1...𝑛))))) |
| 164 | 160, 162,
163 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ran (𝐻 ↾ (𝑓 “ (1...𝑛))) ⊆ (𝑍‘ran (𝐻 ↾ (𝑓 “ (1...𝑛))))) |
| 165 | 97, 50 | eleqtrrdi 2852 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → 𝑛 ∈ ℕ) |
| 166 | | f1ores 6862 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:(1...(♯‘𝑊))–1-1→𝐴 ∧ (1...𝑛) ⊆ (1...(♯‘𝑊))) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1-onto→(𝑓 “ (1...𝑛))) |
| 167 | 147, 101,
166 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1-onto→(𝑓 “ (1...𝑛))) |
| 168 | | f1of1 6847 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1-onto→(𝑓 “ (1...𝑛)) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1→(𝑓 “ (1...𝑛))) |
| 169 | 167, 168 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 ↾ (1...𝑛)):(1...𝑛)–1-1→(𝑓 “ (1...𝑛))) |
| 170 | | suppssdm 8202 |
. . . . . . . . . . . . . . 15
⊢ ((𝐻 ↾ (𝑓 “ (1...𝑛))) supp 0 ) ⊆ dom (𝐻 ↾ (𝑓 “ (1...𝑛))) |
| 171 | | dmres 6030 |
. . . . . . . . . . . . . . . 16
⊢ dom
(𝐻 ↾ (𝑓 “ (1...𝑛))) = ((𝑓 “ (1...𝑛)) ∩ dom 𝐻) |
| 172 | 171 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → dom (𝐻 ↾ (𝑓 “ (1...𝑛))) = ((𝑓 “ (1...𝑛)) ∩ dom 𝐻)) |
| 173 | 170, 172 | sseqtrid 4026 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐻 ↾ (𝑓 “ (1...𝑛))) supp 0 ) ⊆ ((𝑓 “ (1...𝑛)) ∩ dom 𝐻)) |
| 174 | | inss1 4237 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓 “ (1...𝑛)) ∩ dom 𝐻) ⊆ (𝑓 “ (1...𝑛)) |
| 175 | | df-ima 5698 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 “ (1...𝑛)) = ran (𝑓 ↾ (1...𝑛)) |
| 176 | 175 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝑓 “ (1...𝑛)) = ran (𝑓 ↾ (1...𝑛))) |
| 177 | 174, 176 | sseqtrid 4026 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝑓 “ (1...𝑛)) ∩ dom 𝐻) ⊆ ran (𝑓 ↾ (1...𝑛))) |
| 178 | 173, 177 | sstrd 3994 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((𝐻 ↾ (𝑓 “ (1...𝑛))) supp 0 ) ⊆ ran (𝑓 ↾ (1...𝑛))) |
| 179 | | eqid 2737 |
. . . . . . . . . . . . 13
⊢ (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) supp 0 ) = (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) supp 0 ) |
| 180 | 2, 3, 6, 155, 95, 156, 158, 164, 165, 169, 178, 179 | gsumval3 19925 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛)))) = (seq1( + , ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))))‘𝑛)) |
| 181 | 175 | eqimss2i 4045 |
. . . . . . . . . . . . . . . . . 18
⊢ ran
(𝑓 ↾ (1...𝑛)) ⊆ (𝑓 “ (1...𝑛)) |
| 182 | | cores 6269 |
. . . . . . . . . . . . . . . . . 18
⊢ (ran
(𝑓 ↾ (1...𝑛)) ⊆ (𝑓 “ (1...𝑛)) → ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) = (𝐻 ∘ (𝑓 ↾ (1...𝑛)))) |
| 183 | 181, 182 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) = (𝐻 ∘ (𝑓 ↾ (1...𝑛))) |
| 184 | | resco 6270 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐻 ∘ 𝑓) ↾ (1...𝑛)) = (𝐻 ∘ (𝑓 ↾ (1...𝑛))) |
| 185 | 183, 184 | eqtr4i 2768 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))) = ((𝐻 ∘ 𝑓) ↾ (1...𝑛)) |
| 186 | 185 | fveq1i 6907 |
. . . . . . . . . . . . . . 15
⊢ (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛)))‘𝑘) = (((𝐻 ∘ 𝑓) ↾ (1...𝑛))‘𝑘) |
| 187 | | fvres 6925 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (1...𝑛) → (((𝐻 ∘ 𝑓) ↾ (1...𝑛))‘𝑘) = ((𝐻 ∘ 𝑓)‘𝑘)) |
| 188 | 186, 187 | eqtrid 2789 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (1...𝑛) → (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛)))‘𝑘) = ((𝐻 ∘ 𝑓)‘𝑘)) |
| 189 | 188 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) ∧ 𝑘 ∈ (1...𝑛)) → (((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛)))‘𝑘) = ((𝐻 ∘ 𝑓)‘𝑘)) |
| 190 | 97, 189 | seqfveq 14067 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , ((𝐻 ↾ (𝑓 “ (1...𝑛))) ∘ (𝑓 ↾ (1...𝑛))))‘𝑛) = (seq1( + , (𝐻 ∘ 𝑓))‘𝑛)) |
| 191 | 180, 190 | eqtr2d 2778 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐻 ∘ 𝑓))‘𝑛) = (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))) |
| 192 | | fvex 6919 |
. . . . . . . . . . . 12
⊢ (seq1(
+ ,
(𝐻 ∘ 𝑓))‘𝑛) ∈ V |
| 193 | 192 | elsn 4641 |
. . . . . . . . . . 11
⊢ ((seq1(
+ ,
(𝐻 ∘ 𝑓))‘𝑛) ∈ {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))} ↔ (seq1( + , (𝐻 ∘ 𝑓))‘𝑛) = (𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))) |
| 194 | 191, 193 | sylibr 234 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (seq1( + , (𝐻 ∘ 𝑓))‘𝑛) ∈ {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) |
| 195 | 6, 155 | cntzi 19347 |
. . . . . . . . . 10
⊢ ((((𝐹 ∘ 𝑓)‘(𝑛 + 1)) ∈ (𝑍‘{(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) ∧ (seq1( + , (𝐻 ∘ 𝑓))‘𝑛) ∈ {(𝐺 Σg (𝐻 ↾ (𝑓 “ (1...𝑛))))}) → (((𝐹 ∘ 𝑓)‘(𝑛 + 1)) + (seq1( + , (𝐻 ∘ 𝑓))‘𝑛)) = ((seq1( + , (𝐻 ∘ 𝑓))‘𝑛) + ((𝐹 ∘ 𝑓)‘(𝑛 + 1)))) |
| 196 | 154, 194,
195 | syl2anc 584 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (((𝐹 ∘ 𝑓)‘(𝑛 + 1)) + (seq1( + , (𝐻 ∘ 𝑓))‘𝑛)) = ((seq1( + , (𝐻 ∘ 𝑓))‘𝑛) + ((𝐹 ∘ 𝑓)‘(𝑛 + 1)))) |
| 197 | 196 | eqcomd 2743 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → ((seq1( + , (𝐻 ∘ 𝑓))‘𝑛) + ((𝐹 ∘ 𝑓)‘(𝑛 + 1))) = (((𝐹 ∘ 𝑓)‘(𝑛 + 1)) + (seq1( + , (𝐻 ∘ 𝑓))‘𝑛))) |
| 198 | 2, 6, 95, 108, 111, 114, 116, 197 | mnd4g 18761 |
. . . . . . 7
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑛 ∈ (1..^(♯‘𝑊))) → (((seq1( + , (𝐹 ∘ 𝑓))‘𝑛) + (seq1( + , (𝐻 ∘ 𝑓))‘𝑛)) + (((𝐹 ∘ 𝑓)‘(𝑛 + 1)) + ((𝐻 ∘ 𝑓)‘(𝑛 + 1)))) = (((seq1( + , (𝐹 ∘ 𝑓))‘𝑛) + ((𝐹 ∘ 𝑓)‘(𝑛 + 1))) + ((seq1( + , (𝐻 ∘ 𝑓))‘𝑛) + ((𝐻 ∘ 𝑓)‘(𝑛 + 1))))) |
| 199 | 48, 48, 51, 73, 77, 94, 198 | seqcaopr3 14078 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (seq1( + , ((𝐹 ∘f + 𝐻) ∘ 𝑓))‘(♯‘𝑊)) = ((seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) + (seq1( + , (𝐻 ∘ 𝑓))‘(♯‘𝑊)))) |
| 200 | 47, 52, 74, 80, 80, 82 | off 7715 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐹 ∘f + 𝐻):𝐴⟶𝐵) |
| 201 | | gsumzaddlem.3 |
. . . . . . . 8
⊢ (𝜑 → ran (𝐹 ∘f + 𝐻) ⊆ (𝑍‘ran (𝐹 ∘f + 𝐻))) |
| 202 | 201 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran (𝐹 ∘f + 𝐻) ⊆ (𝑍‘ran (𝐹 ∘f + 𝐻))) |
| 203 | 44, 106 | sylan 580 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ (𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑘 + 𝑥) ∈ 𝐵) |
| 204 | 203, 52, 74, 80, 80, 82 | off 7715 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐹 ∘f + 𝐻):𝐴⟶𝐵) |
| 205 | | eldifi 4131 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (𝐴 ∖ ran 𝑓) → 𝑥 ∈ 𝐴) |
| 206 | | eqidd 2738 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) |
| 207 | | eqidd 2738 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = (𝐻‘𝑥)) |
| 208 | 78, 79, 80, 80, 82, 206, 207 | ofval 7708 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f + 𝐻)‘𝑥) = ((𝐹‘𝑥) + (𝐻‘𝑥))) |
| 209 | 205, 208 | sylan2 593 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ((𝐹 ∘f + 𝐻)‘𝑥) = ((𝐹‘𝑥) + (𝐻‘𝑥))) |
| 210 | 16 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐹 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 )) |
| 211 | | f1ofo 6855 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → 𝑓:(1...(♯‘𝑊))–onto→𝑊) |
| 212 | | forn 6823 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:(1...(♯‘𝑊))–onto→𝑊 → ran 𝑓 = 𝑊) |
| 213 | 211, 212 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → ran 𝑓 = 𝑊) |
| 214 | 213, 17 | eqtrdi 2793 |
. . . . . . . . . . . . . 14
⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → ran 𝑓 = ((𝐹 ∪ 𝐻) supp 0 )) |
| 215 | 214 | sseq2d 4016 |
. . . . . . . . . . . . 13
⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 ))) |
| 216 | 215 | ad2antll 729 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 ))) |
| 217 | 210, 216 | mpbird 257 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐹 supp 0 ) ⊆ ran 𝑓) |
| 218 | 12 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 0 ∈ V) |
| 219 | 52, 217, 80, 218 | suppssr 8220 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → (𝐹‘𝑥) = 0 ) |
| 220 | 26 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐻 supp 0 ) ⊆ ((𝐻 ∪ 𝐹) supp 0 )) |
| 221 | 220, 28 | sseqtrrdi 4025 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐻 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 )) |
| 222 | 214 | sseq2d 4016 |
. . . . . . . . . . . . 13
⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → ((𝐻 supp 0 ) ⊆ ran 𝑓 ↔ (𝐻 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 ))) |
| 223 | 222 | ad2antll 729 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ((𝐻 supp 0 ) ⊆ ran 𝑓 ↔ (𝐻 supp 0 ) ⊆ ((𝐹 ∪ 𝐻) supp 0 ))) |
| 224 | 221, 223 | mpbird 257 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐻 supp 0 ) ⊆ ran 𝑓) |
| 225 | 74, 224, 80, 218 | suppssr 8220 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → (𝐻‘𝑥) = 0 ) |
| 226 | 219, 225 | oveq12d 7449 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ((𝐹‘𝑥) + (𝐻‘𝑥)) = ( 0 + 0 )) |
| 227 | 8 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ( 0 + 0 ) = 0 ) |
| 228 | 209, 226,
227 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (𝐴 ∖ ran 𝑓)) → ((𝐹 ∘f + 𝐻)‘𝑥) = 0 ) |
| 229 | 204, 228 | suppss 8219 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ((𝐹 ∘f + 𝐻) supp 0 ) ⊆ ran 𝑓) |
| 230 | | ovex 7464 |
. . . . . . . . 9
⊢ (𝐹 ∘f + 𝐻) ∈ V |
| 231 | 230, 131 | coex 7952 |
. . . . . . . 8
⊢ ((𝐹 ∘f + 𝐻) ∘ 𝑓) ∈ V |
| 232 | | suppimacnv 8199 |
. . . . . . . . 9
⊢ ((((𝐹 ∘f + 𝐻) ∘ 𝑓) ∈ V ∧ 0 ∈ V) → (((𝐹 ∘f + 𝐻) ∘ 𝑓) supp 0 ) = (◡((𝐹 ∘f + 𝐻) ∘ 𝑓) “ (V ∖ { 0 }))) |
| 233 | 232 | eqcomd 2743 |
. . . . . . . 8
⊢ ((((𝐹 ∘f + 𝐻) ∘ 𝑓) ∈ V ∧ 0 ∈ V) → (◡((𝐹 ∘f + 𝐻) ∘ 𝑓) “ (V ∖ { 0 })) = (((𝐹 ∘f + 𝐻) ∘ 𝑓) supp 0 )) |
| 234 | 231, 12, 233 | mp2an 692 |
. . . . . . 7
⊢ (◡((𝐹 ∘f + 𝐻) ∘ 𝑓) “ (V ∖ { 0 })) = (((𝐹 ∘f + 𝐻) ∘ 𝑓) supp 0 ) |
| 235 | 2, 3, 6, 155, 44, 80, 200, 202, 49, 68, 229, 234 | gsumval3 19925 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐺 Σg (𝐹 ∘f + 𝐻)) = (seq1( + , ((𝐹 ∘f + 𝐻) ∘ 𝑓))‘(♯‘𝑊))) |
| 236 | | gsumzaddlem.1 |
. . . . . . . . 9
⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
| 237 | 236 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
| 238 | | eqid 2737 |
. . . . . . . 8
⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 ) |
| 239 | 2, 3, 6, 155, 44, 80, 52, 237, 49, 68, 217, 238 | gsumval3 19925 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐺 Σg 𝐹) = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) |
| 240 | 159 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran 𝐻 ⊆ (𝑍‘ran 𝐻)) |
| 241 | | eqid 2737 |
. . . . . . . 8
⊢ ((𝐻 ∘ 𝑓) supp 0 ) = ((𝐻 ∘ 𝑓) supp 0 ) |
| 242 | 2, 3, 6, 155, 44, 80, 74, 240, 49, 68, 224, 241 | gsumval3 19925 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐺 Σg 𝐻) = (seq1( + , (𝐻 ∘ 𝑓))‘(♯‘𝑊))) |
| 243 | 239, 242 | oveq12d 7449 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)) = ((seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) + (seq1( + , (𝐻 ∘ 𝑓))‘(♯‘𝑊)))) |
| 244 | 199, 235,
243 | 3eqtr4d 2787 |
. . . . 5
⊢ ((𝜑 ∧ ((♯‘𝑊) ∈ ℕ ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |
| 245 | 244 | expr 456 |
. . . 4
⊢ ((𝜑 ∧ (♯‘𝑊) ∈ ℕ) → (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))) |
| 246 | 245 | exlimdv 1933 |
. . 3
⊢ ((𝜑 ∧ (♯‘𝑊) ∈ ℕ) →
(∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))) |
| 247 | 246 | expimpd 453 |
. 2
⊢ (𝜑 → (((♯‘𝑊) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊) → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻)))) |
| 248 | | gsumzadd.fn |
. . . . 5
⊢ (𝜑 → 𝐹 finSupp 0 ) |
| 249 | | gsumzadd.hn |
. . . . 5
⊢ (𝜑 → 𝐻 finSupp 0 ) |
| 250 | 248, 249 | fsuppun 9427 |
. . . 4
⊢ (𝜑 → ((𝐹 ∪ 𝐻) supp 0 ) ∈
Fin) |
| 251 | 17, 250 | eqeltrid 2845 |
. . 3
⊢ (𝜑 → 𝑊 ∈ Fin) |
| 252 | | fz1f1o 15746 |
. . 3
⊢ (𝑊 ∈ Fin → (𝑊 = ∅ ∨
((♯‘𝑊) ∈
ℕ ∧ ∃𝑓
𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊))) |
| 253 | 251, 252 | syl 17 |
. 2
⊢ (𝜑 → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊))) |
| 254 | 43, 247, 253 | mpjaod 861 |
1
⊢ (𝜑 → (𝐺 Σg (𝐹 ∘f + 𝐻)) = ((𝐺 Σg 𝐹) + (𝐺 Σg 𝐻))) |