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Mirrors > Home > MPE Home > Th. List > pws1 | Structured version Visualization version GIF version |
Description: Value of the ring unity in a structure power. (Contributed by Mario Carneiro, 11-Mar-2015.) |
Ref | Expression |
---|---|
pws1.y | β’ π = (π βs πΌ) |
pws1.o | β’ 1 = (1rβπ ) |
Ref | Expression |
---|---|
pws1 | β’ ((π β Ring β§ πΌ β π) β (πΌ Γ { 1 }) = (1rβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pws1.y | . . . 4 β’ π = (π βs πΌ) | |
2 | eqid 2733 | . . . 4 β’ (Scalarβπ ) = (Scalarβπ ) | |
3 | 1, 2 | pwsval 17376 | . . 3 β’ ((π β Ring β§ πΌ β π) β π = ((Scalarβπ )Xs(πΌ Γ {π }))) |
4 | 3 | fveq2d 6850 | . 2 β’ ((π β Ring β§ πΌ β π) β (1rβπ) = (1rβ((Scalarβπ )Xs(πΌ Γ {π })))) |
5 | eqid 2733 | . . 3 β’ ((Scalarβπ )Xs(πΌ Γ {π })) = ((Scalarβπ )Xs(πΌ Γ {π })) | |
6 | simpr 486 | . . 3 β’ ((π β Ring β§ πΌ β π) β πΌ β π) | |
7 | fvexd 6861 | . . 3 β’ ((π β Ring β§ πΌ β π) β (Scalarβπ ) β V) | |
8 | fconst6g 6735 | . . . 4 β’ (π β Ring β (πΌ Γ {π }):πΌβΆRing) | |
9 | 8 | adantr 482 | . . 3 β’ ((π β Ring β§ πΌ β π) β (πΌ Γ {π }):πΌβΆRing) |
10 | 5, 6, 7, 9 | prds1 20046 | . 2 β’ ((π β Ring β§ πΌ β π) β (1r β (πΌ Γ {π })) = (1rβ((Scalarβπ )Xs(πΌ Γ {π })))) |
11 | fn0g 18526 | . . . . . 6 β’ 0g Fn V | |
12 | fnmgp 19906 | . . . . . 6 β’ mulGrp Fn V | |
13 | ssv 3972 | . . . . . . 7 β’ ran mulGrp β V | |
14 | 13 | a1i 11 | . . . . . 6 β’ ((π β Ring β§ πΌ β π) β ran mulGrp β V) |
15 | fnco 6622 | . . . . . 6 β’ ((0g Fn V β§ mulGrp Fn V β§ ran mulGrp β V) β (0g β mulGrp) Fn V) | |
16 | 11, 12, 14, 15 | mp3an12i 1466 | . . . . 5 β’ ((π β Ring β§ πΌ β π) β (0g β mulGrp) Fn V) |
17 | df-ur 19922 | . . . . . 6 β’ 1r = (0g β mulGrp) | |
18 | 17 | fneq1i 6603 | . . . . 5 β’ (1r Fn V β (0g β mulGrp) Fn V) |
19 | 16, 18 | sylibr 233 | . . . 4 β’ ((π β Ring β§ πΌ β π) β 1r Fn V) |
20 | elex 3465 | . . . . 5 β’ (π β Ring β π β V) | |
21 | 20 | adantr 482 | . . . 4 β’ ((π β Ring β§ πΌ β π) β π β V) |
22 | fcoconst 7084 | . . . 4 β’ ((1r Fn V β§ π β V) β (1r β (πΌ Γ {π })) = (πΌ Γ {(1rβπ )})) | |
23 | 19, 21, 22 | syl2anc 585 | . . 3 β’ ((π β Ring β§ πΌ β π) β (1r β (πΌ Γ {π })) = (πΌ Γ {(1rβπ )})) |
24 | pws1.o | . . . . 5 β’ 1 = (1rβπ ) | |
25 | 24 | sneqi 4601 | . . . 4 β’ { 1 } = {(1rβπ )} |
26 | 25 | xpeq2i 5664 | . . 3 β’ (πΌ Γ { 1 }) = (πΌ Γ {(1rβπ )}) |
27 | 23, 26 | eqtr4di 2791 | . 2 β’ ((π β Ring β§ πΌ β π) β (1r β (πΌ Γ {π })) = (πΌ Γ { 1 })) |
28 | 4, 10, 27 | 3eqtr2rd 2780 | 1 β’ ((π β Ring β§ πΌ β π) β (πΌ Γ { 1 }) = (1rβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3447 β wss 3914 {csn 4590 Γ cxp 5635 ran crn 5638 β ccom 5641 Fn wfn 6495 βΆwf 6496 βcfv 6500 (class class class)co 7361 Scalarcsca 17144 0gc0g 17329 Xscprds 17335 βs cpws 17336 mulGrpcmgp 19904 1rcur 19921 Ringcrg 19972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-hom 17165 df-cco 17166 df-0g 17331 df-prds 17337 df-pws 17339 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mgp 19905 df-ur 19922 df-ring 19974 |
This theorem is referenced by: pwspjmhmmgpd 20051 evlsbagval 40795 |
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