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Mirrors > Home > MPE Home > Th. List > pws0g | Structured version Visualization version GIF version |
Description: Zero in a structure power of a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.) |
Ref | Expression |
---|---|
pwsmnd.y | β’ π = (π βs πΌ) |
pws0g.z | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
pws0g | β’ ((π β Mnd β§ πΌ β π) β (πΌ Γ { 0 }) = (0gβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 β’ ((Scalarβπ )Xs(πΌ Γ {π })) = ((Scalarβπ )Xs(πΌ Γ {π })) | |
2 | simpr 484 | . . 3 β’ ((π β Mnd β§ πΌ β π) β πΌ β π) | |
3 | fvexd 6912 | . . 3 β’ ((π β Mnd β§ πΌ β π) β (Scalarβπ ) β V) | |
4 | fconst6g 6786 | . . . 4 β’ (π β Mnd β (πΌ Γ {π }):πΌβΆMnd) | |
5 | 4 | adantr 480 | . . 3 β’ ((π β Mnd β§ πΌ β π) β (πΌ Γ {π }):πΌβΆMnd) |
6 | 1, 2, 3, 5 | prds0g 18727 | . 2 β’ ((π β Mnd β§ πΌ β π) β (0g β (πΌ Γ {π })) = (0gβ((Scalarβπ )Xs(πΌ Γ {π })))) |
7 | fconstmpt 5740 | . . 3 β’ (πΌ Γ { 0 }) = (π₯ β πΌ β¦ 0 ) | |
8 | elex 3490 | . . . . 5 β’ (π β Mnd β π β V) | |
9 | 8 | ad2antrr 725 | . . . 4 β’ (((π β Mnd β§ πΌ β π) β§ π₯ β πΌ) β π β V) |
10 | fconstmpt 5740 | . . . . 5 β’ (πΌ Γ {π }) = (π₯ β πΌ β¦ π ) | |
11 | 10 | a1i 11 | . . . 4 β’ ((π β Mnd β§ πΌ β π) β (πΌ Γ {π }) = (π₯ β πΌ β¦ π )) |
12 | fn0g 18622 | . . . . . 6 β’ 0g Fn V | |
13 | 12 | a1i 11 | . . . . 5 β’ ((π β Mnd β§ πΌ β π) β 0g Fn V) |
14 | dffn5 6957 | . . . . 5 β’ (0g Fn V β 0g = (π β V β¦ (0gβπ))) | |
15 | 13, 14 | sylib 217 | . . . 4 β’ ((π β Mnd β§ πΌ β π) β 0g = (π β V β¦ (0gβπ))) |
16 | fveq2 6897 | . . . . 5 β’ (π = π β (0gβπ) = (0gβπ )) | |
17 | pws0g.z | . . . . 5 β’ 0 = (0gβπ ) | |
18 | 16, 17 | eqtr4di 2786 | . . . 4 β’ (π = π β (0gβπ) = 0 ) |
19 | 9, 11, 15, 18 | fmptco 7138 | . . 3 β’ ((π β Mnd β§ πΌ β π) β (0g β (πΌ Γ {π })) = (π₯ β πΌ β¦ 0 )) |
20 | 7, 19 | eqtr4id 2787 | . 2 β’ ((π β Mnd β§ πΌ β π) β (πΌ Γ { 0 }) = (0g β (πΌ Γ {π }))) |
21 | pwsmnd.y | . . . 4 β’ π = (π βs πΌ) | |
22 | eqid 2728 | . . . 4 β’ (Scalarβπ ) = (Scalarβπ ) | |
23 | 21, 22 | pwsval 17467 | . . 3 β’ ((π β Mnd β§ πΌ β π) β π = ((Scalarβπ )Xs(πΌ Γ {π }))) |
24 | 23 | fveq2d 6901 | . 2 β’ ((π β Mnd β§ πΌ β π) β (0gβπ) = (0gβ((Scalarβπ )Xs(πΌ Γ {π })))) |
25 | 6, 20, 24 | 3eqtr4d 2778 | 1 β’ ((π β Mnd β§ πΌ β π) β (πΌ Γ { 0 }) = (0gβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 Vcvv 3471 {csn 4629 β¦ cmpt 5231 Γ cxp 5676 β ccom 5682 Fn wfn 6543 βΆwf 6544 βcfv 6548 (class class class)co 7420 Scalarcsca 17235 0gc0g 17420 Xscprds 17426 βs cpws 17427 Mndcmnd 18693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 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 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 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-pws 17430 df-mgm 18599 df-sgrp 18678 df-mnd 18694 |
This theorem is referenced by: pwsdiagmhm 18782 pwsco1mhm 18783 pwsco2mhm 18784 frlm0 21687 plypf1 26145 evls1fpws 33246 evlsvvval 41796 pwssplit4 42513 pwslnmlem2 42517 |
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