Step | Hyp | Ref
| Expression |
1 | | gsumval3.b |
. . 3
⊢ 𝐵 = (Base‘𝐺) |
2 | | gsumval3.0 |
. . 3
⊢ 0 =
(0g‘𝐺) |
3 | | gsumval3.p |
. . 3
⊢ + =
(+g‘𝐺) |
4 | | eqid 2737 |
. . 3
⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} |
5 | | gsumval3a.w |
. . . . 5
⊢ 𝑊 = (𝐹 supp 0 ) |
6 | 5 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝑊 = (𝐹 supp 0 )) |
7 | | gsumval3.f |
. . . . . 6
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
8 | | gsumval3.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
9 | 7, 8 | fexd 7043 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ V) |
10 | 2 | fvexi 6731 |
. . . . 5
⊢ 0 ∈
V |
11 | | suppimacnv 7916 |
. . . . 5
⊢ ((𝐹 ∈ V ∧ 0 ∈ V)
→ (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
12 | 9, 10, 11 | sylancl 589 |
. . . 4
⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
13 | | gsumval3.g |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Mnd) |
14 | 1, 2, 3, 4 | gsumvallem2 18260 |
. . . . . . . 8
⊢ (𝐺 ∈ Mnd → {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 }) |
15 | 13, 14 | syl 17 |
. . . . . . 7
⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 }) |
16 | 15 | eqcomd 2743 |
. . . . . 6
⊢ (𝜑 → { 0 } = {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}) |
17 | 16 | difeq2d 4037 |
. . . . 5
⊢ (𝜑 → (V ∖ { 0 }) = (V
∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})) |
18 | 17 | imaeq2d 5929 |
. . . 4
⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) = (◡𝐹 “ (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))) |
19 | 6, 12, 18 | 3eqtrd 2781 |
. . 3
⊢ (𝜑 → 𝑊 = (◡𝐹 “ (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))) |
20 | 1, 2, 3, 4, 19, 13, 8, 7 | gsumval 18149 |
. 2
⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}, 0 , if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))))) |
21 | | gsumval3a.n |
. . . 4
⊢ (𝜑 → 𝑊 ≠ ∅) |
22 | 15 | sseq2d 3933 |
. . . . . 6
⊢ (𝜑 → (ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} ↔ ran 𝐹 ⊆ { 0 })) |
23 | 5 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = (𝐹 supp 0 )) |
24 | 7, 8 | jca 515 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉)) |
25 | 24 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉)) |
26 | | fex 7042 |
. . . . . . . . . 10
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) |
27 | 25, 26 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 ∈ V) |
28 | 27, 10, 11 | sylancl 589 |
. . . . . . . 8
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 }))) |
29 | 7 | ffnd 6546 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 Fn 𝐴) |
30 | 29 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 Fn 𝐴) |
31 | | simpr 488 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → ran 𝐹 ⊆ { 0 }) |
32 | | df-f 6384 |
. . . . . . . . . 10
⊢ (𝐹:𝐴⟶{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ { 0 })) |
33 | 30, 31, 32 | sylanbrc 586 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹:𝐴⟶{ 0 }) |
34 | | disjdif 4386 |
. . . . . . . . 9
⊢ ({ 0 } ∩ (V
∖ { 0 })) =
∅ |
35 | | fimacnvdisj 6597 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶{ 0 } ∧ ({ 0 } ∩ (V
∖ { 0 })) = ∅) →
(◡𝐹 “ (V ∖ { 0 })) =
∅) |
36 | 33, 34, 35 | sylancl 589 |
. . . . . . . 8
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (◡𝐹 “ (V ∖ { 0 })) =
∅) |
37 | 23, 28, 36 | 3eqtrd 2781 |
. . . . . . 7
⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = ∅) |
38 | 37 | ex 416 |
. . . . . 6
⊢ (𝜑 → (ran 𝐹 ⊆ { 0 } → 𝑊 = ∅)) |
39 | 22, 38 | sylbid 243 |
. . . . 5
⊢ (𝜑 → (ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} → 𝑊 = ∅)) |
40 | 39 | necon3ad 2953 |
. . . 4
⊢ (𝜑 → (𝑊 ≠ ∅ → ¬ ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})) |
41 | 21, 40 | mpd 15 |
. . 3
⊢ (𝜑 → ¬ ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}) |
42 | 41 | iffalsed 4450 |
. 2
⊢ (𝜑 → if(ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}, 0 , if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))) = if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))) |
43 | | gsumval3a.i |
. . 3
⊢ (𝜑 → ¬ 𝐴 ∈ ran ...) |
44 | 43 | iffalsed 4450 |
. 2
⊢ (𝜑 → if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))) = (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))) |
45 | 20, 42, 44 | 3eqtrd 2781 |
1
⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))) |