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Mirrors > Home > ILE Home > Th. List > fprod2d | Unicode version |
Description: Write a double product as a product over a two-dimensional region. Compare fsum2d 11398. (Contributed by Scott Fenton, 30-Jan-2018.) |
Ref | Expression |
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fprod2d.1 | |
fprod2d.2 | |
fprod2d.3 | |
fprod2d.4 |
Ref | Expression |
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fprod2d |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3167 | . 2 | |
2 | fprod2d.2 | . . 3 | |
3 | sseq1 3170 | . . . . . 6 | |
4 | prodeq1 11516 | . . . . . . 7 | |
5 | iuneq1 3886 | . . . . . . . . 9 | |
6 | 0iun 3930 | . . . . . . . . 9 | |
7 | 5, 6 | eqtrdi 2219 | . . . . . . . 8 |
8 | 7 | prodeq1d 11527 | . . . . . . 7 |
9 | 4, 8 | eqeq12d 2185 | . . . . . 6 |
10 | 3, 9 | imbi12d 233 | . . . . 5 |
11 | 10 | imbi2d 229 | . . . 4 |
12 | sseq1 3170 | . . . . . 6 | |
13 | prodeq1 11516 | . . . . . . 7 | |
14 | iuneq1 3886 | . . . . . . . 8 | |
15 | 14 | prodeq1d 11527 | . . . . . . 7 |
16 | 13, 15 | eqeq12d 2185 | . . . . . 6 |
17 | 12, 16 | imbi12d 233 | . . . . 5 |
18 | 17 | imbi2d 229 | . . . 4 |
19 | sseq1 3170 | . . . . . 6 | |
20 | prodeq1 11516 | . . . . . . 7 | |
21 | iuneq1 3886 | . . . . . . . 8 | |
22 | 21 | prodeq1d 11527 | . . . . . . 7 |
23 | 20, 22 | eqeq12d 2185 | . . . . . 6 |
24 | 19, 23 | imbi12d 233 | . . . . 5 |
25 | 24 | imbi2d 229 | . . . 4 |
26 | sseq1 3170 | . . . . . 6 | |
27 | prodeq1 11516 | . . . . . . 7 | |
28 | iuneq1 3886 | . . . . . . . 8 | |
29 | 28 | prodeq1d 11527 | . . . . . . 7 |
30 | 27, 29 | eqeq12d 2185 | . . . . . 6 |
31 | 26, 30 | imbi12d 233 | . . . . 5 |
32 | 31 | imbi2d 229 | . . . 4 |
33 | prod0 11548 | . . . . . 6 | |
34 | prod0 11548 | . . . . . 6 | |
35 | 33, 34 | eqtr4i 2194 | . . . . 5 |
36 | 35 | 2a1i 27 | . . . 4 |
37 | ssun1 3290 | . . . . . . . . 9 | |
38 | sstr 3155 | . . . . . . . . 9 | |
39 | 37, 38 | mpan 422 | . . . . . . . 8 |
40 | 39 | imim1i 60 | . . . . . . 7 |
41 | fprod2d.1 | . . . . . . . . . 10 | |
42 | 2 | ad2antrr 485 | . . . . . . . . . 10 |
43 | fprod2d.3 | . . . . . . . . . . 11 | |
44 | 43 | ad4ant14 511 | . . . . . . . . . 10 |
45 | fprod2d.4 | . . . . . . . . . . 11 | |
46 | 45 | ad4ant14 511 | . . . . . . . . . 10 |
47 | simplrr 531 | . . . . . . . . . 10 | |
48 | simpr 109 | . . . . . . . . . 10 | |
49 | simplrl 530 | . . . . . . . . . 10 | |
50 | biid 170 | . . . . . . . . . 10 | |
51 | 41, 42, 44, 46, 47, 48, 49, 50 | fprod2dlemstep 11585 | . . . . . . . . 9 |
52 | 51 | exp31 362 | . . . . . . . 8 |
53 | 52 | a2d 26 | . . . . . . 7 |
54 | 40, 53 | syl5 32 | . . . . . 6 |
55 | 54 | expcom 115 | . . . . 5 |
56 | 55 | a2d 26 | . . . 4 |
57 | 11, 18, 25, 32, 36, 56 | findcard2s 6868 | . . 3 |
58 | 2, 57 | mpcom 36 | . 2 |
59 | 1, 58 | mpi 15 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1348 wcel 2141 cun 3119 wss 3121 c0 3414 csn 3583 cop 3586 ciun 3873 cxp 4609 cfn 6718 cc 7772 c1 7775 cprod 11513 |
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-in1 609 ax-in2 610 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-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 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-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-disj 3967 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-proddc 11514 |
This theorem is referenced by: fprodxp 11587 fprodcom2fi 11589 |
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