![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lmodpropd | Structured version Visualization version GIF version |
Description: If two structures have the same components (properties), one is a left module iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 27-Jun-2015.) |
Ref | Expression |
---|---|
lmodpropd.1 | β’ (π β π΅ = (BaseβπΎ)) |
lmodpropd.2 | β’ (π β π΅ = (BaseβπΏ)) |
lmodpropd.3 | β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) |
lmodpropd.4 | β’ (π β πΉ = (ScalarβπΎ)) |
lmodpropd.5 | β’ (π β πΉ = (ScalarβπΏ)) |
lmodpropd.6 | β’ π = (BaseβπΉ) |
lmodpropd.7 | β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπΏ)π¦)) |
Ref | Expression |
---|---|
lmodpropd | β’ (π β (πΎ β LMod β πΏ β LMod)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodpropd.1 | . 2 β’ (π β π΅ = (BaseβπΎ)) | |
2 | lmodpropd.2 | . 2 β’ (π β π΅ = (BaseβπΏ)) | |
3 | eqid 2733 | . 2 β’ (ScalarβπΎ) = (ScalarβπΎ) | |
4 | eqid 2733 | . 2 β’ (ScalarβπΏ) = (ScalarβπΏ) | |
5 | lmodpropd.6 | . . 3 β’ π = (BaseβπΉ) | |
6 | lmodpropd.4 | . . . 4 β’ (π β πΉ = (ScalarβπΎ)) | |
7 | 6 | fveq2d 6847 | . . 3 β’ (π β (BaseβπΉ) = (Baseβ(ScalarβπΎ))) |
8 | 5, 7 | eqtrid 2785 | . 2 β’ (π β π = (Baseβ(ScalarβπΎ))) |
9 | lmodpropd.5 | . . . 4 β’ (π β πΉ = (ScalarβπΏ)) | |
10 | 9 | fveq2d 6847 | . . 3 β’ (π β (BaseβπΉ) = (Baseβ(ScalarβπΏ))) |
11 | 5, 10 | eqtrid 2785 | . 2 β’ (π β π = (Baseβ(ScalarβπΏ))) |
12 | lmodpropd.3 | . 2 β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) | |
13 | 6, 9 | eqtr3d 2775 | . . . . 5 β’ (π β (ScalarβπΎ) = (ScalarβπΏ)) |
14 | 13 | adantr 482 | . . . 4 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (ScalarβπΎ) = (ScalarβπΏ)) |
15 | 14 | fveq2d 6847 | . . 3 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (+gβ(ScalarβπΎ)) = (+gβ(ScalarβπΏ))) |
16 | 15 | oveqd 7375 | . 2 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯(+gβ(ScalarβπΎ))π¦) = (π₯(+gβ(ScalarβπΏ))π¦)) |
17 | 14 | fveq2d 6847 | . . 3 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (.rβ(ScalarβπΎ)) = (.rβ(ScalarβπΏ))) |
18 | 17 | oveqd 7375 | . 2 β’ ((π β§ (π₯ β π β§ π¦ β π)) β (π₯(.rβ(ScalarβπΎ))π¦) = (π₯(.rβ(ScalarβπΏ))π¦)) |
19 | lmodpropd.7 | . 2 β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπΏ)π¦)) | |
20 | 1, 2, 3, 4, 8, 11, 12, 16, 18, 19 | lmodprop2d 20399 | 1 β’ (π β (πΎ β LMod β πΏ β LMod)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 .rcmulr 17139 Scalarcsca 17141 Β·π cvsca 17142 LModclmod 20336 |
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-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-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-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 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-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 |
This theorem is referenced by: lmhmpropd 20549 lvecpropd 20644 assapropd 21291 opsrlmod 21633 matlmod 21794 mnringlmodd 42594 |
Copyright terms: Public domain | W3C validator |