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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem58 | Structured version Unicode version |
Description: This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 91. Here D is used to represent the set A of Lemma 2, because here the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
stoweidlem58.1 |
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stoweidlem58.2 |
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stoweidlem58.3 |
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stoweidlem58.4 |
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stoweidlem58.5 |
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stoweidlem58.6 |
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stoweidlem58.7 |
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stoweidlem58.8 |
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stoweidlem58.9 |
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stoweidlem58.10 |
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stoweidlem58.11 |
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stoweidlem58.12 |
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stoweidlem58.13 |
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stoweidlem58.14 |
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stoweidlem58.15 |
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stoweidlem58.16 |
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stoweidlem58.17 |
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stoweidlem58.18 |
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Ref | Expression |
---|---|
stoweidlem58 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stoweidlem58.1 |
. . 3
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2 | stoweidlem58.3 |
. . . 4
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3 | 1 | nfeq1 2631 |
. . . 4
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4 | 2, 3 | nfan 1866 |
. . 3
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5 | eqid 2454 |
. . 3
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6 | stoweidlem58.5 |
. . 3
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7 | stoweidlem58.11 |
. . . 4
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8 | 7 | adantlr 714 |
. . 3
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9 | stoweidlem58.13 |
. . . 4
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10 | 9 | adantr 465 |
. . 3
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11 | stoweidlem58.17 |
. . . 4
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12 | 11 | adantr 465 |
. . 3
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13 | simpr 461 |
. . 3
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14 | 1, 4, 5, 6, 8, 10, 12, 13 | stoweidlem18 29984 |
. 2
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15 | stoweidlem58.2 |
. . 3
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16 | nfcv 2616 |
. . . . 5
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17 | 1, 16 | nfne 2783 |
. . . 4
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18 | 2, 17 | nfan 1866 |
. . 3
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19 | eqid 2454 |
. . 3
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20 | eqid 2454 |
. . 3
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21 | stoweidlem58.4 |
. . 3
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22 | stoweidlem58.6 |
. . 3
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23 | stoweidlem58.16 |
. . 3
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24 | stoweidlem58.7 |
. . . 4
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25 | 24 | adantr 465 |
. . 3
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26 | stoweidlem58.8 |
. . . 4
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27 | 26 | adantr 465 |
. . 3
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28 | stoweidlem58.9 |
. . . 4
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29 | 28 | 3adant1r 1212 |
. . 3
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30 | stoweidlem58.10 |
. . . 4
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31 | 30 | 3adant1r 1212 |
. . 3
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32 | 7 | adantlr 714 |
. . 3
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33 | stoweidlem58.12 |
. . . 4
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34 | 33 | adantlr 714 |
. . 3
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35 | 9 | adantr 465 |
. . 3
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36 | stoweidlem58.14 |
. . . 4
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37 | 36 | adantr 465 |
. . 3
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38 | stoweidlem58.15 |
. . . 4
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39 | 38 | adantr 465 |
. . 3
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40 | simpr 461 |
. . 3
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41 | 11 | adantr 465 |
. . 3
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42 | stoweidlem58.18 |
. . . 4
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43 | 42 | adantr 465 |
. . 3
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44 | 1, 15, 18, 19, 20, 21, 6, 22, 23, 25, 27, 29, 31, 32, 34, 35, 37, 39, 40, 41, 43 | stoweidlem57 30023 |
. 2
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45 | 14, 44 | pm2.61dane 2770 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-inf2 7962 ax-cnex 9453 ax-resscn 9454 ax-1cn 9455 ax-icn 9456 ax-addcl 9457 ax-addrcl 9458 ax-mulcl 9459 ax-mulrcl 9460 ax-mulcom 9461 ax-addass 9462 ax-mulass 9463 ax-distr 9464 ax-i2m1 9465 ax-1ne0 9466 ax-1rid 9467 ax-rnegex 9468 ax-rrecex 9469 ax-cnre 9470 ax-pre-lttri 9471 ax-pre-lttrn 9472 ax-pre-ltadd 9473 ax-pre-mulgt0 9474 ax-pre-sup 9475 ax-mulf 9477 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-fal 1376 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-int 4240 df-iun 4284 df-iin 4285 df-br 4404 df-opab 4462 df-mpt 4463 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-se 4791 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-isom 5538 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-of 6433 df-om 6590 df-1st 6690 df-2nd 6691 df-supp 6804 df-recs 6945 df-rdg 6979 df-1o 7033 df-2o 7034 df-oadd 7037 df-er 7214 df-map 7329 df-pm 7330 df-ixp 7377 df-en 7424 df-dom 7425 df-sdom 7426 df-fin 7427 df-fsupp 7735 df-fi 7776 df-sup 7806 df-oi 7839 df-card 8224 df-cda 8452 df-pnf 9535 df-mnf 9536 df-xr 9537 df-ltxr 9538 df-le 9539 df-sub 9712 df-neg 9713 df-div 10109 df-nn 10438 df-2 10495 df-3 10496 df-4 10497 df-5 10498 df-6 10499 df-7 10500 df-8 10501 df-9 10502 df-10 10503 df-n0 10695 df-z 10762 df-dec 10871 df-uz 10977 df-q 11069 df-rp 11107 df-xneg 11204 df-xadd 11205 df-xmul 11206 df-ioo 11419 df-ico 11421 df-icc 11422 df-fz 11559 df-fzo 11670 df-fl 11763 df-seq 11928 df-exp 11987 df-hash 12225 df-cj 12710 df-re 12711 df-im 12712 df-sqr 12846 df-abs 12847 df-clim 13088 df-rlim 13089 df-sum 13286 df-struct 14298 df-ndx 14299 df-slot 14300 df-base 14301 df-sets 14302 df-ress 14303 df-plusg 14374 df-mulr 14375 df-starv 14376 df-sca 14377 df-vsca 14378 df-ip 14379 df-tset 14380 df-ple 14381 df-ds 14383 df-unif 14384 df-hom 14385 df-cco 14386 df-rest 14484 df-topn 14485 df-0g 14503 df-gsum 14504 df-topgen 14505 df-pt 14506 df-prds 14509 df-xrs 14563 df-qtop 14568 df-imas 14569 df-xps 14571 df-mre 14647 df-mrc 14648 df-acs 14650 df-mnd 15538 df-submnd 15588 df-mulg 15671 df-cntz 15958 df-cmn 16404 df-psmet 17944 df-xmet 17945 df-met 17946 df-bl 17947 df-mopn 17948 df-cnfld 17954 df-top 18645 df-bases 18647 df-topon 18648 df-topsp 18649 df-cld 18765 df-cn 18973 df-cnp 18974 df-cmp 19132 df-tx 19277 df-hmeo 19470 df-xms 20037 df-ms 20038 df-tms 20039 |
This theorem is referenced by: stoweidlem59 30025 |
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