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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 |
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prodfdivap.2 |
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prodfdivap.3 |
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prodfdivap.4 |
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prodfdivap.5 |
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Ref | Expression |
---|---|
prodfdivap |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodfdiv.1 |
. . . 4
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2 | prodfdivap.3 |
. . . 4
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3 | elfzuz 10040 |
. . . . 5
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4 | prodfdivap.4 |
. . . . 5
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5 | 3, 4 | sylan2 286 |
. . . 4
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6 | eqid 2189 |
. . . . . 6
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7 | fveq2 5530 |
. . . . . . 7
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8 | 7 | oveq2d 5907 |
. . . . . 6
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9 | simpr 110 |
. . . . . 6
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10 | 2, 4 | recclapd 8757 |
. . . . . 6
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11 | 6, 8, 9, 10 | fvmptd3 5625 |
. . . . 5
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12 | 3, 11 | sylan2 286 |
. . . 4
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13 | 11, 10 | eqeltrd 2266 |
. . . 4
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14 | 1, 2, 5, 12, 13 | prodfrecap 11573 |
. . 3
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15 | 14 | oveq2d 5907 |
. 2
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16 | prodfdivap.2 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | eleq1w 2250 |
. . . . . . . . 9
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18 | 17 | anbi2d 464 |
. . . . . . . 8
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19 | fveq2 5530 |
. . . . . . . . 9
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20 | 19 | eleq1d 2258 |
. . . . . . . 8
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21 | 18, 20 | imbi12d 234 |
. . . . . . 7
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22 | 21, 2 | chvarvv 1920 |
. . . . . 6
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23 | 19 | breq1d 4028 |
. . . . . . . 8
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24 | 18, 23 | imbi12d 234 |
. . . . . . 7
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25 | 24, 4 | chvarvv 1920 |
. . . . . 6
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26 | 22, 25 | recclapd 8757 |
. . . . 5
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27 | 26 | fmpttd 5687 |
. . . 4
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28 | 27 | ffvelcdmda 5667 |
. . 3
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29 | 16, 2, 4 | divrecapd 8769 |
. . . 4
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30 | prodfdivap.5 |
. . . 4
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31 | 11 | oveq2d 5907 |
. . . 4
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32 | 29, 30, 31 | 3eqtr4d 2232 |
. . 3
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33 | 1, 16, 28, 32 | prod3fmul 11568 |
. 2
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34 | eqid 2189 |
. . . . 5
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35 | eluzel2 9552 |
. . . . . 6
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36 | 1, 35 | syl 14 |
. . . . 5
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37 | 34, 36, 16 | prodf 11565 |
. . . 4
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38 | 37, 1 | ffvelcdmd 5668 |
. . 3
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39 | 34, 36, 2 | prodf 11565 |
. . . 4
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40 | 39, 1 | ffvelcdmd 5668 |
. . 3
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41 | 1, 2, 5 | prodfap0 11572 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 38, 40, 41 | divrecapd 8769 |
. 2
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43 | 15, 33, 42 | 3eqtr4d 2232 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-pre-mulext 7948 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-frec 6410 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-ap 8558 df-div 8649 df-inn 8939 df-n0 9196 df-z 9273 df-uz 9548 df-fz 10028 df-fzo 10162 df-seqfrec 10465 |
This theorem is referenced by: (None) |
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