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Mirrors > Home > ILE Home > Th. List > gausslemma2dlem5 | Unicode version |
Description: Lemma 5 for gausslemma2d 15133. (Contributed by AV, 9-Jul-2021.) |
Ref | Expression |
---|---|
gausslemma2d.p |
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gausslemma2d.h |
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gausslemma2d.r |
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gausslemma2d.m |
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gausslemma2d.n |
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Ref | Expression |
---|---|
gausslemma2dlem5 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gausslemma2d.p |
. . 3
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2 | gausslemma2d.h |
. . 3
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3 | gausslemma2d.r |
. . 3
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4 | gausslemma2d.m |
. . 3
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5 | 1, 2, 3, 4 | gausslemma2dlem5a 15129 |
. 2
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6 | 1 | gausslemma2dlem0a 15113 |
. . . . . . . . . . 11
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7 | 6 | nnzd 9428 |
. . . . . . . . . 10
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8 | 4nn 9135 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() | |
9 | znq 9679 |
. . . . . . . . . 10
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10 | 7, 8, 9 | sylancl 413 |
. . . . . . . . 9
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11 | 10 | flqcld 10336 |
. . . . . . . 8
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12 | 4, 11 | eqeltrid 2280 |
. . . . . . 7
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13 | 12 | peano2zd 9432 |
. . . . . 6
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14 | 1, 2 | gausslemma2dlem0b 15114 |
. . . . . . 7
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15 | 14 | nnzd 9428 |
. . . . . 6
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16 | 13, 15 | fzfigd 10492 |
. . . . 5
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17 | neg1cn 9077 |
. . . . . 6
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18 | 17 | a1i 9 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | elfzelz 10081 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 2z 9335 |
. . . . . . . . 9
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21 | 20 | a1i 9 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 19, 21 | zmulcld 9435 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | zcnd 9430 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 23 | adantl 277 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 16, 18, 24 | fprodmul 11724 |
. . . 4
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26 | fprodconst 11753 |
. . . . . . 7
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27 | 16, 17, 26 | sylancl 413 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | nnoddn2prm 12388 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | nnz 9326 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | oddm1d2 12023 |
. . . . . . . . . . . . . 14
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31 | 29, 30 | syl 14 |
. . . . . . . . . . . . 13
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32 | 31 | biimpa 296 |
. . . . . . . . . . . 12
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33 | 1, 28, 32 | 3syl 17 |
. . . . . . . . . . 11
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34 | 2, 33 | eqeltrid 2280 |
. . . . . . . . . 10
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35 | 1, 4, 2 | gausslemma2dlem0f 15118 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | eluz2 9588 |
. . . . . . . . . 10
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37 | 13, 34, 35, 36 | syl3anbrc 1183 |
. . . . . . . . 9
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38 | hashfz 10882 |
. . . . . . . . 9
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39 | 37, 38 | syl 14 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 34 | zcnd 9430 |
. . . . . . . . . 10
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41 | 12 | zcnd 9430 |
. . . . . . . . . 10
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42 | 1cnd 8025 |
. . . . . . . . . 10
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43 | 40, 41, 42 | nppcan2d 8346 |
. . . . . . . . 9
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44 | gausslemma2d.n |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
45 | 43, 44 | eqtr4di 2244 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | 39, 45 | eqtrd 2226 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 46 | oveq2d 5926 |
. . . . . 6
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48 | 27, 47 | eqtrd 2226 |
. . . . 5
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49 | 48 | oveq1d 5925 |
. . . 4
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50 | 25, 49 | eqtrd 2226 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
51 | 50 | oveq1d 5925 |
. 2
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52 | 5, 51 | eqtrd 2226 |
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 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-iinf 4616 ax-cnex 7953 ax-resscn 7954 ax-1cn 7955 ax-1re 7956 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-mulrcl 7961 ax-addcom 7962 ax-mulcom 7963 ax-addass 7964 ax-mulass 7965 ax-distr 7966 ax-i2m1 7967 ax-0lt1 7968 ax-1rid 7969 ax-0id 7970 ax-rnegex 7971 ax-precex 7972 ax-cnre 7973 ax-pre-ltirr 7974 ax-pre-ltwlin 7975 ax-pre-lttrn 7976 ax-pre-apti 7977 ax-pre-ltadd 7978 ax-pre-mulgt0 7979 ax-pre-mulext 7980 ax-arch 7981 ax-caucvg 7982 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-tp 3626 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4322 df-po 4325 df-iso 4326 df-iord 4395 df-on 4397 df-ilim 4398 df-suc 4400 df-iom 4619 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-isom 5255 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-1st 6184 df-2nd 6185 df-recs 6349 df-irdg 6414 df-frec 6435 df-1o 6460 df-2o 6461 df-oadd 6464 df-er 6578 df-en 6786 df-dom 6787 df-fin 6788 df-pnf 8046 df-mnf 8047 df-xr 8048 df-ltxr 8049 df-le 8050 df-sub 8182 df-neg 8183 df-reap 8584 df-ap 8591 df-div 8682 df-inn 8973 df-2 9031 df-3 9032 df-4 9033 df-5 9034 df-6 9035 df-n0 9231 df-z 9308 df-uz 9583 df-q 9675 df-rp 9710 df-fz 10065 df-fzo 10199 df-fl 10329 df-mod 10384 df-seqfrec 10509 df-exp 10600 df-ihash 10837 df-cj 10976 df-re 10977 df-im 10978 df-rsqrt 11132 df-abs 11133 df-clim 11412 df-proddc 11684 df-dvds 11921 df-prm 12236 |
This theorem is referenced by: gausslemma2dlem6 15131 |
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