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Mirrors > Home > MPE Home > Th. List > cvsmuleqdivd | Structured version Visualization version GIF version |
Description: An equality involving ratios in a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.) |
Ref | Expression |
---|---|
cvsdiveqd.v | β’ π = (Baseβπ) |
cvsdiveqd.t | β’ Β· = ( Β·π βπ) |
cvsdiveqd.f | β’ πΉ = (Scalarβπ) |
cvsdiveqd.k | β’ πΎ = (BaseβπΉ) |
cvsdiveqd.w | β’ (π β π β βVec) |
cvsdiveqd.a | β’ (π β π΄ β πΎ) |
cvsdiveqd.b | β’ (π β π΅ β πΎ) |
cvsdiveqd.x | β’ (π β π β π) |
cvsdiveqd.y | β’ (π β π β π) |
cvsdiveqd.1 | β’ (π β π΄ β 0) |
cvsmuleqdivd.1 | β’ (π β (π΄ Β· π) = (π΅ Β· π)) |
Ref | Expression |
---|---|
cvsmuleqdivd | β’ (π β π = ((π΅ / π΄) Β· π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvsmuleqdivd.1 | . . 3 β’ (π β (π΄ Β· π) = (π΅ Β· π)) | |
2 | 1 | oveq2d 7377 | . 2 β’ (π β ((1 / π΄) Β· (π΄ Β· π)) = ((1 / π΄) Β· (π΅ Β· π))) |
3 | cvsdiveqd.w | . . . . . . . 8 β’ (π β π β βVec) | |
4 | 3 | cvsclm 24512 | . . . . . . 7 β’ (π β π β βMod) |
5 | cvsdiveqd.f | . . . . . . . 8 β’ πΉ = (Scalarβπ) | |
6 | cvsdiveqd.k | . . . . . . . 8 β’ πΎ = (BaseβπΉ) | |
7 | 5, 6 | clmsscn 24465 | . . . . . . 7 β’ (π β βMod β πΎ β β) |
8 | 4, 7 | syl 17 | . . . . . 6 β’ (π β πΎ β β) |
9 | cvsdiveqd.a | . . . . . 6 β’ (π β π΄ β πΎ) | |
10 | 8, 9 | sseldd 3949 | . . . . 5 β’ (π β π΄ β β) |
11 | cvsdiveqd.1 | . . . . 5 β’ (π β π΄ β 0) | |
12 | 10, 11 | recid2d 11935 | . . . 4 β’ (π β ((1 / π΄) Β· π΄) = 1) |
13 | 12 | oveq1d 7376 | . . 3 β’ (π β (((1 / π΄) Β· π΄) Β· π) = (1 Β· π)) |
14 | 5 | clm1 24459 | . . . . . . 7 β’ (π β βMod β 1 = (1rβπΉ)) |
15 | 4, 14 | syl 17 | . . . . . 6 β’ (π β 1 = (1rβπΉ)) |
16 | 5 | clmring 24456 | . . . . . . 7 β’ (π β βMod β πΉ β Ring) |
17 | eqid 2733 | . . . . . . . 8 β’ (1rβπΉ) = (1rβπΉ) | |
18 | 6, 17 | ringidcl 19997 | . . . . . . 7 β’ (πΉ β Ring β (1rβπΉ) β πΎ) |
19 | 4, 16, 18 | 3syl 18 | . . . . . 6 β’ (π β (1rβπΉ) β πΎ) |
20 | 15, 19 | eqeltrd 2834 | . . . . 5 β’ (π β 1 β πΎ) |
21 | 5, 6 | cvsdivcl 24519 | . . . . 5 β’ ((π β βVec β§ (1 β πΎ β§ π΄ β πΎ β§ π΄ β 0)) β (1 / π΄) β πΎ) |
22 | 3, 20, 9, 11, 21 | syl13anc 1373 | . . . 4 β’ (π β (1 / π΄) β πΎ) |
23 | cvsdiveqd.x | . . . 4 β’ (π β π β π) | |
24 | cvsdiveqd.v | . . . . 5 β’ π = (Baseβπ) | |
25 | cvsdiveqd.t | . . . . 5 β’ Β· = ( Β·π βπ) | |
26 | 24, 5, 25, 6 | clmvsass 24475 | . . . 4 β’ ((π β βMod β§ ((1 / π΄) β πΎ β§ π΄ β πΎ β§ π β π)) β (((1 / π΄) Β· π΄) Β· π) = ((1 / π΄) Β· (π΄ Β· π))) |
27 | 4, 22, 9, 23, 26 | syl13anc 1373 | . . 3 β’ (π β (((1 / π΄) Β· π΄) Β· π) = ((1 / π΄) Β· (π΄ Β· π))) |
28 | 24, 25 | clmvs1 24479 | . . . 4 β’ ((π β βMod β§ π β π) β (1 Β· π) = π) |
29 | 4, 23, 28 | syl2anc 585 | . . 3 β’ (π β (1 Β· π) = π) |
30 | 13, 27, 29 | 3eqtr3d 2781 | . 2 β’ (π β ((1 / π΄) Β· (π΄ Β· π)) = π) |
31 | cvsdiveqd.b | . . . . . 6 β’ (π β π΅ β πΎ) | |
32 | 8, 31 | sseldd 3949 | . . . . 5 β’ (π β π΅ β β) |
33 | 32, 10, 11 | divrec2d 11943 | . . . 4 β’ (π β (π΅ / π΄) = ((1 / π΄) Β· π΅)) |
34 | 33 | oveq1d 7376 | . . 3 β’ (π β ((π΅ / π΄) Β· π) = (((1 / π΄) Β· π΅) Β· π)) |
35 | cvsdiveqd.y | . . . 4 β’ (π β π β π) | |
36 | 24, 5, 25, 6 | clmvsass 24475 | . . . 4 β’ ((π β βMod β§ ((1 / π΄) β πΎ β§ π΅ β πΎ β§ π β π)) β (((1 / π΄) Β· π΅) Β· π) = ((1 / π΄) Β· (π΅ Β· π))) |
37 | 4, 22, 31, 35, 36 | syl13anc 1373 | . . 3 β’ (π β (((1 / π΄) Β· π΅) Β· π) = ((1 / π΄) Β· (π΅ Β· π))) |
38 | 34, 37 | eqtr2d 2774 | . 2 β’ (π β ((1 / π΄) Β· (π΅ Β· π)) = ((π΅ / π΄) Β· π)) |
39 | 2, 30, 38 | 3eqtr3d 2781 | 1 β’ (π β π = ((π΅ / π΄) Β· π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wne 2940 β wss 3914 βcfv 6500 (class class class)co 7361 βcc 11057 0cc0 11059 1c1 11060 Β· cmul 11064 / cdiv 11820 Basecbs 17091 Scalarcsca 17144 Β·π cvsca 17145 1rcur 19921 Ringcrg 19972 βModcclm 24448 βVecccvs 24509 |
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 ax-addf 11138 ax-mulf 11139 |
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-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 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-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-subg 18933 df-cmn 19572 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-drng 20221 df-subrg 20262 df-lmod 20367 df-lvec 20608 df-cnfld 20820 df-clm 24449 df-cvs 24510 |
This theorem is referenced by: ttgcontlem1 27882 |
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