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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0mulgnn0 | Structured version Visualization version GIF version |
Description: The group multiple function in the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
Ref | Expression |
---|---|
xrge0mulgnn0 | โข ((๐ด โ โ0 โง ๐ต โ (0[,]+โ)) โ (๐ด(.gโ(โ*๐ โพs (0[,]+โ)))๐ต) = (๐ด ยทe ๐ต)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . 3 โข (โ*๐ โพs (0[,]+โ)) = (โ*๐ โพs (0[,]+โ)) | |
2 | iccssxr 13434 | . . . 4 โข (0[,]+โ) โ โ* | |
3 | xrsbas 21310 | . . . 4 โข โ* = (Baseโโ*๐ ) | |
4 | 2, 3 | sseqtri 4010 | . . 3 โข (0[,]+โ) โ (Baseโโ*๐ ) |
5 | eqid 2725 | . . 3 โข (.gโโ*๐ ) = (.gโโ*๐ ) | |
6 | eqid 2725 | . . 3 โข (invgโโ*๐ ) = (invgโโ*๐ ) | |
7 | xrs0 32773 | . . . 4 โข 0 = (0gโโ*๐ ) | |
8 | xrge00 32782 | . . . 4 โข 0 = (0gโ(โ*๐ โพs (0[,]+โ))) | |
9 | 7, 8 | eqtr3i 2755 | . . 3 โข (0gโโ*๐ ) = (0gโ(โ*๐ โพs (0[,]+โ))) |
10 | 1, 4, 5, 6, 9 | ressmulgnn0 19032 | . 2 โข ((๐ด โ โ0 โง ๐ต โ (0[,]+โ)) โ (๐ด(.gโ(โ*๐ โพs (0[,]+โ)))๐ต) = (๐ด(.gโโ*๐ )๐ต)) |
11 | nn0z 12608 | . . 3 โข (๐ด โ โ0 โ ๐ด โ โค) | |
12 | eliccxr 13439 | . . 3 โข (๐ต โ (0[,]+โ) โ ๐ต โ โ*) | |
13 | xrsmulgzz 32776 | . . 3 โข ((๐ด โ โค โง ๐ต โ โ*) โ (๐ด(.gโโ*๐ )๐ต) = (๐ด ยทe ๐ต)) | |
14 | 11, 12, 13 | syl2an 594 | . 2 โข ((๐ด โ โ0 โง ๐ต โ (0[,]+โ)) โ (๐ด(.gโโ*๐ )๐ต) = (๐ด ยทe ๐ต)) |
15 | 10, 14 | eqtrd 2765 | 1 โข ((๐ด โ โ0 โง ๐ต โ (0[,]+โ)) โ (๐ด(.gโ(โ*๐ โพs (0[,]+โ)))๐ต) = (๐ด ยทe ๐ต)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 394 = wceq 1533 โ wcel 2098 โcfv 6543 (class class class)co 7413 0cc0 11133 +โcpnf 11270 โ*cxr 11272 โ0cn0 12497 โคcz 12583 ยทe cxmu 13118 [,]cicc 13354 Basecbs 17174 โพs cress 17203 0gc0g 17415 โ*๐ cxrs 17476 invgcminusg 18890 .gcmg 19022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-icc 13358 df-fz 13512 df-seq 13994 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-tset 17246 df-ple 17247 df-ds 17249 df-0g 17417 df-xrs 17478 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-minusg 18893 df-mulg 19023 df-cmn 19736 |
This theorem is referenced by: esumcst 33735 |
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