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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 2821 | . 2 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾) | |
4 | eqid 2821 | . 2 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿) | |
5 | lmodpropd.6 | . . 3 ⊢ 𝑃 = (Base‘𝐹) | |
6 | lmodpropd.4 | . . . 4 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐾)) | |
7 | 6 | fveq2d 6674 | . . 3 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐾))) |
8 | 5, 7 | syl5eq 2868 | . 2 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾))) |
9 | lmodpropd.5 | . . . 4 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿)) | |
10 | 9 | fveq2d 6674 | . . 3 ⊢ (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐿))) |
11 | 5, 10 | syl5eq 2868 | . 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 2858 | . . . . 5 ⊢ (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿)) |
14 | 13 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (Scalar‘𝐾) = (Scalar‘𝐿)) |
15 | 14 | fveq2d 6674 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (+g‘(Scalar‘𝐾)) = (+g‘(Scalar‘𝐿))) |
16 | 15 | oveqd 7173 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘(Scalar‘𝐾))𝑦) = (𝑥(+g‘(Scalar‘𝐿))𝑦)) |
17 | 14 | fveq2d 6674 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (.r‘(Scalar‘𝐾)) = (.r‘(Scalar‘𝐿))) |
18 | 17 | oveqd 7173 | . 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 19696 | 1 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 Scalarcsca 16568 ·𝑠 cvsca 16569 LModclmod 19634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-mgp 19240 df-ur 19252 df-ring 19299 df-lmod 19636 |
This theorem is referenced by: lmhmpropd 19845 lvecpropd 19939 assapropd 20101 opsrlmod 20414 matlmod 21038 |
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