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Mirrors > Home > MPE Home > Th. List > prdsinvgd | Structured version Visualization version GIF version |
Description: Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
Ref | Expression |
---|---|
prdsgrpd.y | β’ π = (πXsπ ) |
prdsgrpd.i | β’ (π β πΌ β π) |
prdsgrpd.s | β’ (π β π β π) |
prdsgrpd.r | β’ (π β π :πΌβΆGrp) |
prdsinvgd.b | β’ π΅ = (Baseβπ) |
prdsinvgd.n | β’ π = (invgβπ) |
prdsinvgd.x | β’ (π β π β π΅) |
Ref | Expression |
---|---|
prdsinvgd | β’ (π β (πβπ) = (π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsgrpd.y | . . . . 5 β’ π = (πXsπ ) | |
2 | prdsinvgd.b | . . . . 5 β’ π΅ = (Baseβπ) | |
3 | eqid 2725 | . . . . 5 β’ (+gβπ) = (+gβπ) | |
4 | prdsgrpd.s | . . . . . 6 β’ (π β π β π) | |
5 | 4 | elexd 3484 | . . . . 5 β’ (π β π β V) |
6 | prdsgrpd.i | . . . . . 6 β’ (π β πΌ β π) | |
7 | 6 | elexd 3484 | . . . . 5 β’ (π β πΌ β V) |
8 | prdsgrpd.r | . . . . 5 β’ (π β π :πΌβΆGrp) | |
9 | prdsinvgd.x | . . . . 5 β’ (π β π β π΅) | |
10 | eqid 2725 | . . . . 5 β’ (0g β π ) = (0g β π ) | |
11 | eqid 2725 | . . . . 5 β’ (π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯))) = (π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯))) | |
12 | 1, 2, 3, 5, 7, 8, 9, 10, 11 | prdsinvlem 19009 | . . . 4 β’ (π β ((π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯))) β π΅ β§ ((π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯)))(+gβπ)π) = (0g β π ))) |
13 | 12 | simprd 494 | . . 3 β’ (π β ((π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯)))(+gβπ)π) = (0g β π )) |
14 | grpmnd 18901 | . . . . . 6 β’ (π β Grp β π β Mnd) | |
15 | 14 | ssriv 3976 | . . . . 5 β’ Grp β Mnd |
16 | fss 6734 | . . . . 5 β’ ((π :πΌβΆGrp β§ Grp β Mnd) β π :πΌβΆMnd) | |
17 | 8, 15, 16 | sylancl 584 | . . . 4 β’ (π β π :πΌβΆMnd) |
18 | 1, 6, 4, 17 | prds0g 18727 | . . 3 β’ (π β (0g β π ) = (0gβπ)) |
19 | 13, 18 | eqtrd 2765 | . 2 β’ (π β ((π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯)))(+gβπ)π) = (0gβπ)) |
20 | 1, 6, 4, 8 | prdsgrpd 19010 | . . 3 β’ (π β π β Grp) |
21 | 12 | simpld 493 | . . 3 β’ (π β (π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯))) β π΅) |
22 | eqid 2725 | . . . 4 β’ (0gβπ) = (0gβπ) | |
23 | prdsinvgd.n | . . . 4 β’ π = (invgβπ) | |
24 | 2, 3, 22, 23 | grpinvid2 18953 | . . 3 β’ ((π β Grp β§ π β π΅ β§ (π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯))) β π΅) β ((πβπ) = (π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯))) β ((π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯)))(+gβπ)π) = (0gβπ))) |
25 | 20, 9, 21, 24 | syl3anc 1368 | . 2 β’ (π β ((πβπ) = (π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯))) β ((π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯)))(+gβπ)π) = (0gβπ))) |
26 | 19, 25 | mpbird 256 | 1 β’ (π β (πβπ) = (π₯ β πΌ β¦ ((invgβ(π βπ₯))β(πβπ₯)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 Vcvv 3463 β wss 3939 β¦ cmpt 5226 β ccom 5676 βΆwf 6539 βcfv 6543 (class class class)co 7416 Basecbs 17179 +gcplusg 17232 0gc0g 17420 Xscprds 17426 Mndcmnd 18693 Grpcgrp 18894 invgcminusg 18895 |
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-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-prds 17428 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 |
This theorem is referenced by: pwsinvg 19013 prdsinvgd2 21680 prdstgpd 24047 |
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