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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 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
lmodpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
lmodpropd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
lmodpropd.4 | ⊢ (𝜑 → 𝐹 = (Scalar‘𝐾)) |
lmodpropd.5 | ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿)) |
lmodpropd.6 | ⊢ 𝑃 = (Base‘𝐹) |
lmodpropd.7 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) |
Ref | Expression |
---|---|
lmodpropd | ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodpropd.1 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | lmodpropd.2 | . 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
3 | eqid 2738 | . 2 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾) | |
4 | eqid 2738 | . 2 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿) | |
5 | lmodpropd.6 | . . 3 ⊢ 𝑃 = (Base‘𝐹) | |
6 | lmodpropd.4 | . . . 4 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐾)) | |
7 | 6 | fveq2d 6772 | . . 3 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐾))) |
8 | 5, 7 | eqtrid 2790 | . 2 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾))) |
9 | lmodpropd.5 | . . . 4 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿)) | |
10 | 9 | fveq2d 6772 | . . 3 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿))) |
11 | 5, 10 | eqtrid 2790 | . 2 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿))) |
12 | lmodpropd.3 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
13 | 6, 9 | eqtr3d 2780 | . . . . 5 ⊢ (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿)) |
14 | 13 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (Scalar‘𝐾) = (Scalar‘𝐿)) |
15 | 14 | fveq2d 6772 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (+g‘(Scalar‘𝐾)) = (+g‘(Scalar‘𝐿))) |
16 | 15 | oveqd 7286 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘(Scalar‘𝐾))𝑦) = (𝑥(+g‘(Scalar‘𝐿))𝑦)) |
17 | 14 | fveq2d 6772 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (.r‘(Scalar‘𝐾)) = (.r‘(Scalar‘𝐿))) |
18 | 17 | oveqd 7286 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(.r‘(Scalar‘𝐾))𝑦) = (𝑥(.r‘(Scalar‘𝐿))𝑦)) |
19 | lmodpropd.7 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) | |
20 | 1, 2, 3, 4, 8, 11, 12, 16, 18, 19 | lmodprop2d 20174 | 1 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ‘cfv 6428 (class class class)co 7269 Basecbs 16901 +gcplusg 16951 .rcmulr 16952 Scalarcsca 16954 ·𝑠 cvsca 16955 LModclmod 20112 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8487 df-en 8723 df-dom 8724 df-sdom 8725 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-nn 11963 df-2 12025 df-sets 16854 df-slot 16872 df-ndx 16884 df-base 16902 df-plusg 16964 df-0g 17141 df-mgm 18315 df-sgrp 18364 df-mnd 18375 df-grp 18569 df-mgp 19710 df-ur 19727 df-ring 19774 df-lmod 20114 |
This theorem is referenced by: lmhmpropd 20324 lvecpropd 20418 assapropd 21065 opsrlmod 21406 matlmod 21567 mnringlmodd 41804 |
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