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Mirrors > Home > MPE Home > Th. List > frlmvplusgscavalb | Structured version Visualization version GIF version |
Description: Addition combined with scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.) |
Ref | Expression |
---|---|
frlmplusgvalb.f | β’ πΉ = (π freeLMod πΌ) |
frlmplusgvalb.b | β’ π΅ = (BaseβπΉ) |
frlmplusgvalb.i | β’ (π β πΌ β π) |
frlmplusgvalb.x | β’ (π β π β π΅) |
frlmplusgvalb.z | β’ (π β π β π΅) |
frlmplusgvalb.r | β’ (π β π β Ring) |
frlmvscavalb.k | β’ πΎ = (Baseβπ ) |
frlmvscavalb.a | β’ (π β π΄ β πΎ) |
frlmvscavalb.v | β’ β = ( Β·π βπΉ) |
frlmvscavalb.t | β’ Β· = (.rβπ ) |
frlmvplusgscavalb.y | β’ (π β π β π΅) |
frlmvplusgscavalb.a | β’ + = (+gβπ ) |
frlmvplusgscavalb.p | β’ β = (+gβπΉ) |
frlmvplusgscavalb.c | β’ (π β πΆ β πΎ) |
Ref | Expression |
---|---|
frlmvplusgscavalb | β’ (π β (π = ((π΄ β π) β (πΆ β π)) β βπ β πΌ (πβπ) = ((π΄ Β· (πβπ)) + (πΆ Β· (πβπ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmplusgvalb.f | . . 3 β’ πΉ = (π freeLMod πΌ) | |
2 | frlmplusgvalb.b | . . 3 β’ π΅ = (BaseβπΉ) | |
3 | frlmplusgvalb.i | . . 3 β’ (π β πΌ β π) | |
4 | frlmplusgvalb.r | . . . . 5 β’ (π β π β Ring) | |
5 | 1 | frlmlmod 21171 | . . . . 5 β’ ((π β Ring β§ πΌ β π) β πΉ β LMod) |
6 | 4, 3, 5 | syl2anc 585 | . . . 4 β’ (π β πΉ β LMod) |
7 | frlmvscavalb.a | . . . . . 6 β’ (π β π΄ β πΎ) | |
8 | frlmvscavalb.k | . . . . . 6 β’ πΎ = (Baseβπ ) | |
9 | 7, 8 | eleqtrdi 2844 | . . . . 5 β’ (π β π΄ β (Baseβπ )) |
10 | 1 | frlmsca 21175 | . . . . . . 7 β’ ((π β Ring β§ πΌ β π) β π = (ScalarβπΉ)) |
11 | 4, 3, 10 | syl2anc 585 | . . . . . 6 β’ (π β π = (ScalarβπΉ)) |
12 | 11 | fveq2d 6847 | . . . . 5 β’ (π β (Baseβπ ) = (Baseβ(ScalarβπΉ))) |
13 | 9, 12 | eleqtrd 2836 | . . . 4 β’ (π β π΄ β (Baseβ(ScalarβπΉ))) |
14 | frlmplusgvalb.x | . . . 4 β’ (π β π β π΅) | |
15 | eqid 2733 | . . . . 5 β’ (ScalarβπΉ) = (ScalarβπΉ) | |
16 | frlmvscavalb.v | . . . . 5 β’ β = ( Β·π βπΉ) | |
17 | eqid 2733 | . . . . 5 β’ (Baseβ(ScalarβπΉ)) = (Baseβ(ScalarβπΉ)) | |
18 | 2, 15, 16, 17 | lmodvscl 20354 | . . . 4 β’ ((πΉ β LMod β§ π΄ β (Baseβ(ScalarβπΉ)) β§ π β π΅) β (π΄ β π) β π΅) |
19 | 6, 13, 14, 18 | syl3anc 1372 | . . 3 β’ (π β (π΄ β π) β π΅) |
20 | frlmplusgvalb.z | . . 3 β’ (π β π β π΅) | |
21 | frlmvplusgscavalb.c | . . . . . 6 β’ (π β πΆ β πΎ) | |
22 | 21, 8 | eleqtrdi 2844 | . . . . 5 β’ (π β πΆ β (Baseβπ )) |
23 | 22, 12 | eleqtrd 2836 | . . . 4 β’ (π β πΆ β (Baseβ(ScalarβπΉ))) |
24 | frlmvplusgscavalb.y | . . . 4 β’ (π β π β π΅) | |
25 | 2, 15, 16, 17 | lmodvscl 20354 | . . . 4 β’ ((πΉ β LMod β§ πΆ β (Baseβ(ScalarβπΉ)) β§ π β π΅) β (πΆ β π) β π΅) |
26 | 6, 23, 24, 25 | syl3anc 1372 | . . 3 β’ (π β (πΆ β π) β π΅) |
27 | frlmvplusgscavalb.a | . . 3 β’ + = (+gβπ ) | |
28 | frlmvplusgscavalb.p | . . 3 β’ β = (+gβπΉ) | |
29 | 1, 2, 3, 19, 20, 4, 26, 27, 28 | frlmplusgvalb 21191 | . 2 β’ (π β (π = ((π΄ β π) β (πΆ β π)) β βπ β πΌ (πβπ) = (((π΄ β π)βπ) + ((πΆ β π)βπ)))) |
30 | 3 | adantr 482 | . . . . . 6 β’ ((π β§ π β πΌ) β πΌ β π) |
31 | 7 | adantr 482 | . . . . . 6 β’ ((π β§ π β πΌ) β π΄ β πΎ) |
32 | 14 | adantr 482 | . . . . . 6 β’ ((π β§ π β πΌ) β π β π΅) |
33 | simpr 486 | . . . . . 6 β’ ((π β§ π β πΌ) β π β πΌ) | |
34 | frlmvscavalb.t | . . . . . 6 β’ Β· = (.rβπ ) | |
35 | 1, 2, 8, 30, 31, 32, 33, 16, 34 | frlmvscaval 21190 | . . . . 5 β’ ((π β§ π β πΌ) β ((π΄ β π)βπ) = (π΄ Β· (πβπ))) |
36 | 21 | adantr 482 | . . . . . 6 β’ ((π β§ π β πΌ) β πΆ β πΎ) |
37 | 24 | adantr 482 | . . . . . 6 β’ ((π β§ π β πΌ) β π β π΅) |
38 | 1, 2, 8, 30, 36, 37, 33, 16, 34 | frlmvscaval 21190 | . . . . 5 β’ ((π β§ π β πΌ) β ((πΆ β π)βπ) = (πΆ Β· (πβπ))) |
39 | 35, 38 | oveq12d 7376 | . . . 4 β’ ((π β§ π β πΌ) β (((π΄ β π)βπ) + ((πΆ β π)βπ)) = ((π΄ Β· (πβπ)) + (πΆ Β· (πβπ)))) |
40 | 39 | eqeq2d 2744 | . . 3 β’ ((π β§ π β πΌ) β ((πβπ) = (((π΄ β π)βπ) + ((πΆ β π)βπ)) β (πβπ) = ((π΄ Β· (πβπ)) + (πΆ Β· (πβπ))))) |
41 | 40 | ralbidva 3169 | . 2 β’ (π β (βπ β πΌ (πβπ) = (((π΄ β π)βπ) + ((πΆ β π)βπ)) β βπ β πΌ (πβπ) = ((π΄ Β· (πβπ)) + (πΆ Β· (πβπ))))) |
42 | 29, 41 | bitrd 279 | 1 β’ (π β (π = ((π΄ β π) β (πΆ β π)) β βπ β πΌ (πβπ) = ((π΄ Β· (πβπ)) + (πΆ Β· (πβπ))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 .rcmulr 17139 Scalarcsca 17141 Β·π cvsca 17142 Ringcrg 19969 LModclmod 20336 freeLMod cfrlm 21168 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-sup 9383 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-hom 17162 df-cco 17163 df-0g 17328 df-prds 17334 df-pws 17336 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-mgp 19902 df-ur 19919 df-ring 19971 df-subrg 20234 df-lmod 20338 df-lss 20408 df-sra 20649 df-rgmod 20650 df-dsmm 21154 df-frlm 21169 |
This theorem is referenced by: rrxplusgvscavalb 24775 |
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