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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 7421 | . 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 25008 | . . . . . . 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 24961 | . . . . . . 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 3978 | . . . . 5 β’ (π β π΄ β β) |
11 | cvsdiveqd.1 | . . . . 5 β’ (π β π΄ β 0) | |
12 | 10, 11 | recid2d 11990 | . . . 4 β’ (π β ((1 / π΄) Β· π΄) = 1) |
13 | 12 | oveq1d 7420 | . . 3 β’ (π β (((1 / π΄) Β· π΄) Β· π) = (1 Β· π)) |
14 | 5 | clm1 24955 | . . . . . . 7 β’ (π β βMod β 1 = (1rβπΉ)) |
15 | 4, 14 | syl 17 | . . . . . 6 β’ (π β 1 = (1rβπΉ)) |
16 | 5 | clmring 24952 | . . . . . . 7 β’ (π β βMod β πΉ β Ring) |
17 | eqid 2726 | . . . . . . . 8 β’ (1rβπΉ) = (1rβπΉ) | |
18 | 6, 17 | ringidcl 20165 | . . . . . . 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 2827 | . . . . 5 β’ (π β 1 β πΎ) |
21 | 5, 6 | cvsdivcl 25015 | . . . . 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 1369 | . . . 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 24971 | . . . 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 1369 | . . 3 β’ (π β (((1 / π΄) Β· π΄) Β· π) = ((1 / π΄) Β· (π΄ Β· π))) |
28 | 24, 25 | clmvs1 24975 | . . . 4 β’ ((π β βMod β§ π β π) β (1 Β· π) = π) |
29 | 4, 23, 28 | syl2anc 583 | . . 3 β’ (π β (1 Β· π) = π) |
30 | 13, 27, 29 | 3eqtr3d 2774 | . 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 3978 | . . . . 5 β’ (π β π΅ β β) |
33 | 32, 10, 11 | divrec2d 11998 | . . . 4 β’ (π β (π΅ / π΄) = ((1 / π΄) Β· π΅)) |
34 | 33 | oveq1d 7420 | . . 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 24971 | . . . 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 1369 | . . 3 β’ (π β (((1 / π΄) Β· π΅) Β· π) = ((1 / π΄) Β· (π΅ Β· π))) |
38 | 34, 37 | eqtr2d 2767 | . 2 β’ (π β ((1 / π΄) Β· (π΅ Β· π)) = ((π΅ / π΄) Β· π)) |
39 | 2, 30, 38 | 3eqtr3d 2774 | 1 β’ (π β π = ((π΅ / π΄) Β· π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wne 2934 β wss 3943 βcfv 6537 (class class class)co 7405 βcc 11110 0cc0 11112 1c1 11113 Β· cmul 11117 / cdiv 11875 Basecbs 17153 Scalarcsca 17209 Β·π cvsca 17210 1rcur 20086 Ringcrg 20138 βModcclm 24944 βVecccvs 25005 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-subg 19050 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-subrg 20471 df-drng 20589 df-lmod 20708 df-lvec 20951 df-cnfld 21241 df-clm 24945 df-cvs 25006 |
This theorem is referenced by: ttgcontlem1 28650 |
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