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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 11336. (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 3148 | . 2 | |
2 | fprod2d.2 | . . 3 | |
3 | sseq1 3151 | . . . . . 6 | |
4 | prodeq1 11454 | . . . . . . 7 | |
5 | iuneq1 3863 | . . . . . . . . 9 | |
6 | 0iun 3907 | . . . . . . . . 9 | |
7 | 5, 6 | eqtrdi 2206 | . . . . . . . 8 |
8 | 7 | prodeq1d 11465 | . . . . . . 7 |
9 | 4, 8 | eqeq12d 2172 | . . . . . 6 |
10 | 3, 9 | imbi12d 233 | . . . . 5 |
11 | 10 | imbi2d 229 | . . . 4 |
12 | sseq1 3151 | . . . . . 6 | |
13 | prodeq1 11454 | . . . . . . 7 | |
14 | iuneq1 3863 | . . . . . . . 8 | |
15 | 14 | prodeq1d 11465 | . . . . . . 7 |
16 | 13, 15 | eqeq12d 2172 | . . . . . 6 |
17 | 12, 16 | imbi12d 233 | . . . . 5 |
18 | 17 | imbi2d 229 | . . . 4 |
19 | sseq1 3151 | . . . . . 6 | |
20 | prodeq1 11454 | . . . . . . 7 | |
21 | iuneq1 3863 | . . . . . . . 8 | |
22 | 21 | prodeq1d 11465 | . . . . . . 7 |
23 | 20, 22 | eqeq12d 2172 | . . . . . 6 |
24 | 19, 23 | imbi12d 233 | . . . . 5 |
25 | 24 | imbi2d 229 | . . . 4 |
26 | sseq1 3151 | . . . . . 6 | |
27 | prodeq1 11454 | . . . . . . 7 | |
28 | iuneq1 3863 | . . . . . . . 8 | |
29 | 28 | prodeq1d 11465 | . . . . . . 7 |
30 | 27, 29 | eqeq12d 2172 | . . . . . 6 |
31 | 26, 30 | imbi12d 233 | . . . . 5 |
32 | 31 | imbi2d 229 | . . . 4 |
33 | prod0 11486 | . . . . . 6 | |
34 | prod0 11486 | . . . . . 6 | |
35 | 33, 34 | eqtr4i 2181 | . . . . 5 |
36 | 35 | 2a1i 27 | . . . 4 |
37 | ssun1 3270 | . . . . . . . . 9 | |
38 | sstr 3136 | . . . . . . . . 9 | |
39 | 37, 38 | mpan 421 | . . . . . . . 8 |
40 | 39 | imim1i 60 | . . . . . . 7 |
41 | fprod2d.1 | . . . . . . . . . 10 | |
42 | 2 | ad2antrr 480 | . . . . . . . . . 10 |
43 | fprod2d.3 | . . . . . . . . . . 11 | |
44 | 43 | ad4ant14 506 | . . . . . . . . . 10 |
45 | fprod2d.4 | . . . . . . . . . . 11 | |
46 | 45 | ad4ant14 506 | . . . . . . . . . 10 |
47 | simplrr 526 | . . . . . . . . . 10 | |
48 | simpr 109 | . . . . . . . . . 10 | |
49 | simplrl 525 | . . . . . . . . . 10 | |
50 | biid 170 | . . . . . . . . . 10 | |
51 | 41, 42, 44, 46, 47, 48, 49, 50 | fprod2dlemstep 11523 | . . . . . . . . 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 6836 | . . 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 1335 wcel 2128 cun 3100 wss 3102 c0 3394 csn 3560 cop 3563 ciun 3850 cxp 4585 cfn 6686 cc 7731 c1 7734 cprod 11451 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 ax-arch 7852 ax-caucvg 7853 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-disj 3944 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-isom 5180 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-irdg 6318 df-frec 6339 df-1o 6364 df-oadd 6368 df-er 6481 df-en 6687 df-dom 6688 df-fin 6689 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 df-inn 8835 df-2 8893 df-3 8894 df-4 8895 df-n0 9092 df-z 9169 df-uz 9441 df-q 9530 df-rp 9562 df-fz 9914 df-fzo 10046 df-seqfrec 10349 df-exp 10423 df-ihash 10654 df-cj 10746 df-re 10747 df-im 10748 df-rsqrt 10902 df-abs 10903 df-clim 11180 df-proddc 11452 |
This theorem is referenced by: fprodxp 11525 fprodcom2fi 11527 |
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