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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 2622 | . 2 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾) | |
4 | eqid 2622 | . 2 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿) | |
5 | lmodpropd.6 | . . 3 ⊢ 𝑃 = (Base‘𝐹) | |
6 | lmodpropd.4 | . . . 4 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐾)) | |
7 | 6 | fveq2d 6195 | . . 3 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐾))) |
8 | 5, 7 | syl5eq 2668 | . 2 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾))) |
9 | lmodpropd.5 | . . . 4 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿)) | |
10 | 9 | fveq2d 6195 | . . 3 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿))) |
11 | 5, 10 | syl5eq 2668 | . 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 2658 | . . . . 5 ⊢ (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿)) |
14 | 13 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (Scalar‘𝐾) = (Scalar‘𝐿)) |
15 | 14 | fveq2d 6195 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (+g‘(Scalar‘𝐾)) = (+g‘(Scalar‘𝐿))) |
16 | 15 | oveqd 6667 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘(Scalar‘𝐾))𝑦) = (𝑥(+g‘(Scalar‘𝐿))𝑦)) |
17 | 14 | fveq2d 6195 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (.r‘(Scalar‘𝐾)) = (.r‘(Scalar‘𝐿))) |
18 | 17 | oveqd 6667 | . 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 18925 | 1 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 Scalarcsca 15944 ·𝑠 cvsca 15945 LModclmod 18863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-mgp 18490 df-ur 18502 df-ring 18549 df-lmod 18865 |
This theorem is referenced by: lmhmpropd 19073 lvecpropd 19167 assapropd 19327 opsrlmod 19616 matlmod 20235 |
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