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Mirrors > Home > ILE Home > Th. List > prodfdivap | Unicode version |
Description: The quotient of two products. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.) |
Ref | Expression |
---|---|
prodfdiv.1 | |
prodfdivap.2 | |
prodfdivap.3 | |
prodfdivap.4 | # |
prodfdivap.5 |
Ref | Expression |
---|---|
prodfdivap |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodfdiv.1 | . . . 4 | |
2 | prodfdivap.3 | . . . 4 | |
3 | elfzuz 9833 | . . . . 5 | |
4 | prodfdivap.4 | . . . . 5 # | |
5 | 3, 4 | sylan2 284 | . . . 4 # |
6 | eqid 2140 | . . . . . 6 | |
7 | fveq2 5429 | . . . . . . 7 | |
8 | 7 | oveq2d 5798 | . . . . . 6 |
9 | simpr 109 | . . . . . 6 | |
10 | 2, 4 | recclapd 8565 | . . . . . 6 |
11 | 6, 8, 9, 10 | fvmptd3 5522 | . . . . 5 |
12 | 3, 11 | sylan2 284 | . . . 4 |
13 | 11, 10 | eqeltrd 2217 | . . . 4 |
14 | 1, 2, 5, 12, 13 | prodfrecap 11347 | . . 3 |
15 | 14 | oveq2d 5798 | . 2 |
16 | prodfdivap.2 | . . 3 | |
17 | eleq1w 2201 | . . . . . . . . 9 | |
18 | 17 | anbi2d 460 | . . . . . . . 8 |
19 | fveq2 5429 | . . . . . . . . 9 | |
20 | 19 | eleq1d 2209 | . . . . . . . 8 |
21 | 18, 20 | imbi12d 233 | . . . . . . 7 |
22 | 21, 2 | chvarvv 1881 | . . . . . 6 |
23 | 19 | breq1d 3947 | . . . . . . . 8 # # |
24 | 18, 23 | imbi12d 233 | . . . . . . 7 # # |
25 | 24, 4 | chvarvv 1881 | . . . . . 6 # |
26 | 22, 25 | recclapd 8565 | . . . . 5 |
27 | 26 | fmpttd 5583 | . . . 4 |
28 | 27 | ffvelrnda 5563 | . . 3 |
29 | 16, 2, 4 | divrecapd 8577 | . . . 4 |
30 | prodfdivap.5 | . . . 4 | |
31 | 11 | oveq2d 5798 | . . . 4 |
32 | 29, 30, 31 | 3eqtr4d 2183 | . . 3 |
33 | 1, 16, 28, 32 | prod3fmul 11342 | . 2 |
34 | eqid 2140 | . . . . 5 | |
35 | eluzel2 9355 | . . . . . 6 | |
36 | 1, 35 | syl 14 | . . . . 5 |
37 | 34, 36, 16 | prodf 11339 | . . . 4 |
38 | 37, 1 | ffvelrnd 5564 | . . 3 |
39 | 34, 36, 2 | prodf 11339 | . . . 4 |
40 | 39, 1 | ffvelrnd 5564 | . . 3 |
41 | 1, 2, 5 | prodfap0 11346 | . . 3 # |
42 | 38, 40, 41 | divrecapd 8577 | . 2 |
43 | 15, 33, 42 | 3eqtr4d 2183 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1332 wcel 1481 class class class wbr 3937 cmpt 3997 cfv 5131 (class class class)co 5782 cc 7642 cc0 7644 c1 7645 cmul 7649 # cap 8367 cdiv 8456 cz 9078 cuz 9350 cfz 9821 cseq 10249 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-fz 9822 df-fzo 9951 df-seqfrec 10250 |
This theorem is referenced by: (None) |
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