Step | Hyp | Ref
| Expression |
1 | | gsumval3.b |
. . 3
β’ π΅ = (BaseβπΊ) |
2 | | gsumval3.0 |
. . 3
β’ 0 =
(0gβπΊ) |
3 | | gsumval3.p |
. . 3
β’ + =
(+gβπΊ) |
4 | | eqid 2733 |
. . 3
β’ {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)} = {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)} |
5 | | gsumval3a.w |
. . . . 5
β’ π = (πΉ supp 0 ) |
6 | 5 | a1i 11 |
. . . 4
β’ (π β π = (πΉ supp 0 )) |
7 | | gsumval3.f |
. . . . . 6
β’ (π β πΉ:π΄βΆπ΅) |
8 | | gsumval3.a |
. . . . . 6
β’ (π β π΄ β π) |
9 | 7, 8 | fexd 7181 |
. . . . 5
β’ (π β πΉ β V) |
10 | 2 | fvexi 6860 |
. . . . 5
β’ 0 β
V |
11 | | suppimacnv 8109 |
. . . . 5
β’ ((πΉ β V β§ 0 β V)
β (πΉ supp 0 ) = (β‘πΉ β (V β { 0 }))) |
12 | 9, 10, 11 | sylancl 587 |
. . . 4
β’ (π β (πΉ supp 0 ) = (β‘πΉ β (V β { 0 }))) |
13 | | gsumval3.g |
. . . . . . . 8
β’ (π β πΊ β Mnd) |
14 | 1, 2, 3, 4 | gsumvallem2 18652 |
. . . . . . . 8
β’ (πΊ β Mnd β {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)} = { 0 }) |
15 | 13, 14 | syl 17 |
. . . . . . 7
β’ (π β {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)} = { 0 }) |
16 | 15 | eqcomd 2739 |
. . . . . 6
β’ (π β { 0 } = {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)}) |
17 | 16 | difeq2d 4086 |
. . . . 5
β’ (π β (V β { 0 }) = (V
β {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)})) |
18 | 17 | imaeq2d 6017 |
. . . 4
β’ (π β (β‘πΉ β (V β { 0 })) = (β‘πΉ β (V β {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)}))) |
19 | 6, 12, 18 | 3eqtrd 2777 |
. . 3
β’ (π β π = (β‘πΉ β (V β {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)}))) |
20 | 1, 2, 3, 4, 19, 13, 8, 7 | gsumval 18540 |
. 2
β’ (π β (πΊ Ξ£g πΉ) = if(ran πΉ β {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)}, 0 , if(π΄ β ran ..., (β©π₯βπβπ β (β€β₯βπ)(π΄ = (π...π) β§ π₯ = (seqπ( + , πΉ)βπ))), (β©π₯βπ(π:(1...(β―βπ))β1-1-ontoβπ β§ π₯ = (seq1( + , (πΉ β π))β(β―βπ))))))) |
21 | | gsumval3a.n |
. . . 4
β’ (π β π β β
) |
22 | 15 | sseq2d 3980 |
. . . . . 6
β’ (π β (ran πΉ β {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)} β ran πΉ β { 0 })) |
23 | 5 | a1i 11 |
. . . . . . . 8
β’ ((π β§ ran πΉ β { 0 }) β π = (πΉ supp 0 )) |
24 | 7, 8 | jca 513 |
. . . . . . . . . . 11
β’ (π β (πΉ:π΄βΆπ΅ β§ π΄ β π)) |
25 | 24 | adantr 482 |
. . . . . . . . . 10
β’ ((π β§ ran πΉ β { 0 }) β (πΉ:π΄βΆπ΅ β§ π΄ β π)) |
26 | | fex 7180 |
. . . . . . . . . 10
β’ ((πΉ:π΄βΆπ΅ β§ π΄ β π) β πΉ β V) |
27 | 25, 26 | syl 17 |
. . . . . . . . 9
β’ ((π β§ ran πΉ β { 0 }) β πΉ β V) |
28 | 27, 10, 11 | sylancl 587 |
. . . . . . . 8
β’ ((π β§ ran πΉ β { 0 }) β (πΉ supp 0 ) = (β‘πΉ β (V β { 0 }))) |
29 | 7 | ffnd 6673 |
. . . . . . . . . . 11
β’ (π β πΉ Fn π΄) |
30 | 29 | adantr 482 |
. . . . . . . . . 10
β’ ((π β§ ran πΉ β { 0 }) β πΉ Fn π΄) |
31 | | simpr 486 |
. . . . . . . . . 10
β’ ((π β§ ran πΉ β { 0 }) β ran πΉ β { 0 }) |
32 | | df-f 6504 |
. . . . . . . . . 10
β’ (πΉ:π΄βΆ{ 0 } β (πΉ Fn π΄ β§ ran πΉ β { 0 })) |
33 | 30, 31, 32 | sylanbrc 584 |
. . . . . . . . 9
β’ ((π β§ ran πΉ β { 0 }) β πΉ:π΄βΆ{ 0 }) |
34 | | disjdif 4435 |
. . . . . . . . 9
β’ ({ 0 } β© (V
β { 0 })) =
β
|
35 | | fimacnvdisj 6724 |
. . . . . . . . 9
β’ ((πΉ:π΄βΆ{ 0 } β§ ({ 0 } β© (V
β { 0 })) = β
) β
(β‘πΉ β (V β { 0 })) =
β
) |
36 | 33, 34, 35 | sylancl 587 |
. . . . . . . 8
β’ ((π β§ ran πΉ β { 0 }) β (β‘πΉ β (V β { 0 })) =
β
) |
37 | 23, 28, 36 | 3eqtrd 2777 |
. . . . . . 7
β’ ((π β§ ran πΉ β { 0 }) β π = β
) |
38 | 37 | ex 414 |
. . . . . 6
β’ (π β (ran πΉ β { 0 } β π = β
)) |
39 | 22, 38 | sylbid 239 |
. . . . 5
β’ (π β (ran πΉ β {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)} β π = β
)) |
40 | 39 | necon3ad 2953 |
. . . 4
β’ (π β (π β β
β Β¬ ran πΉ β {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)})) |
41 | 21, 40 | mpd 15 |
. . 3
β’ (π β Β¬ ran πΉ β {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)}) |
42 | 41 | iffalsed 4501 |
. 2
β’ (π β if(ran πΉ β {π§ β π΅ β£ βπ¦ β π΅ ((π§ + π¦) = π¦ β§ (π¦ + π§) = π¦)}, 0 , if(π΄ β ran ..., (β©π₯βπβπ β (β€β₯βπ)(π΄ = (π...π) β§ π₯ = (seqπ( + , πΉ)βπ))), (β©π₯βπ(π:(1...(β―βπ))β1-1-ontoβπ β§ π₯ = (seq1( + , (πΉ β π))β(β―βπ)))))) = if(π΄ β ran ..., (β©π₯βπβπ β (β€β₯βπ)(π΄ = (π...π) β§ π₯ = (seqπ( + , πΉ)βπ))), (β©π₯βπ(π:(1...(β―βπ))β1-1-ontoβπ β§ π₯ = (seq1( + , (πΉ β π))β(β―βπ)))))) |
43 | | gsumval3a.i |
. . 3
β’ (π β Β¬ π΄ β ran ...) |
44 | 43 | iffalsed 4501 |
. 2
β’ (π β if(π΄ β ran ..., (β©π₯βπβπ β (β€β₯βπ)(π΄ = (π...π) β§ π₯ = (seqπ( + , πΉ)βπ))), (β©π₯βπ(π:(1...(β―βπ))β1-1-ontoβπ β§ π₯ = (seq1( + , (πΉ β π))β(β―βπ))))) = (β©π₯βπ(π:(1...(β―βπ))β1-1-ontoβπ β§ π₯ = (seq1( + , (πΉ β π))β(β―βπ))))) |
45 | 20, 42, 44 | 3eqtrd 2777 |
1
β’ (π β (πΊ Ξ£g πΉ) = (β©π₯βπ(π:(1...(β―βπ))β1-1-ontoβπ β§ π₯ = (seq1( + , (πΉ β π))β(β―βπ))))) |