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Mirrors > Home > ILE Home > Th. List > fsumgcl | Unicode version |
Description: Closure for a function used to describe a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.) |
Ref | Expression |
---|---|
fsum.1 | |
fsum.2 | |
fsum.3 | |
fsum.4 | |
fsum.5 |
Ref | Expression |
---|---|
fsumgcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsum.5 | . . 3 | |
2 | fsum.3 | . . . . . . 7 | |
3 | f1of 5440 | . . . . . . 7 | |
4 | 2, 3 | syl 14 | . . . . . 6 |
5 | 4 | ffvelrnda 5629 | . . . . 5 |
6 | fsum.1 | . . . . . 6 | |
7 | 6 | adantl 275 | . . . . 5 |
8 | 5, 7 | csbied 3095 | . . . 4 |
9 | fsum.4 | . . . . . . 7 | |
10 | 9 | ralrimiva 2543 | . . . . . 6 |
11 | 10 | adantr 274 | . . . . 5 |
12 | nfcsb1v 3082 | . . . . . . 7 | |
13 | 12 | nfel1 2323 | . . . . . 6 |
14 | csbeq1a 3058 | . . . . . . 7 | |
15 | 14 | eleq1d 2239 | . . . . . 6 |
16 | 13, 15 | rspc 2828 | . . . . 5 |
17 | 5, 11, 16 | sylc 62 | . . . 4 |
18 | 8, 17 | eqeltrrd 2248 | . . 3 |
19 | 1, 18 | eqeltrd 2247 | . 2 |
20 | 19 | ralrimiva 2543 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 wral 2448 csb 3049 wf 5192 wf1o 5195 cfv 5196 (class class class)co 5851 cc 7765 c1 7768 cn 8871 cfz 9958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-f1o 5203 df-fv 5204 |
This theorem is referenced by: fsum3 11343 fprodseq 11539 |
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