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Mirrors > Home > MPE Home > Th. List > lvecprop2d | Structured version Visualization version GIF version |
Description: If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 18988 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.) |
Ref | Expression |
---|---|
lvecprop2d.b1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
lvecprop2d.b2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
lvecprop2d.f | ⊢ 𝐹 = (Scalar‘𝐾) |
lvecprop2d.g | ⊢ 𝐺 = (Scalar‘𝐿) |
lvecprop2d.p1 | ⊢ (𝜑 → 𝑃 = (Base‘𝐹)) |
lvecprop2d.p2 | ⊢ (𝜑 → 𝑃 = (Base‘𝐺)) |
lvecprop2d.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
lvecprop2d.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦)) |
lvecprop2d.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(.r‘𝐹)𝑦) = (𝑥(.r‘𝐺)𝑦)) |
lvecprop2d.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) |
Ref | Expression |
---|---|
lvecprop2d | ⊢ (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecprop2d.b1 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | lvecprop2d.b2 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
3 | lvecprop2d.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝐾) | |
4 | lvecprop2d.g | . . . 4 ⊢ 𝐺 = (Scalar‘𝐿) | |
5 | lvecprop2d.p1 | . . . 4 ⊢ (𝜑 → 𝑃 = (Base‘𝐹)) | |
6 | lvecprop2d.p2 | . . . 4 ⊢ (𝜑 → 𝑃 = (Base‘𝐺)) | |
7 | lvecprop2d.1 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
8 | lvecprop2d.2 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦)) | |
9 | lvecprop2d.3 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(.r‘𝐹)𝑦) = (𝑥(.r‘𝐺)𝑦)) | |
10 | lvecprop2d.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | lmodprop2d 18748 | . . 3 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod)) |
12 | 5, 6, 8, 9 | drngpropd 18597 | . . 3 ⊢ (𝜑 → (𝐹 ∈ DivRing ↔ 𝐺 ∈ DivRing)) |
13 | 11, 12 | anbi12d 743 | . 2 ⊢ (𝜑 → ((𝐾 ∈ LMod ∧ 𝐹 ∈ DivRing) ↔ (𝐿 ∈ LMod ∧ 𝐺 ∈ DivRing))) |
14 | 3 | islvec 18925 | . 2 ⊢ (𝐾 ∈ LVec ↔ (𝐾 ∈ LMod ∧ 𝐹 ∈ DivRing)) |
15 | 4 | islvec 18925 | . 2 ⊢ (𝐿 ∈ LVec ↔ (𝐿 ∈ LMod ∧ 𝐺 ∈ DivRing)) |
16 | 13, 14, 15 | 3bitr4g 302 | 1 ⊢ (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 Scalarcsca 15771 ·𝑠 cvsca 15772 DivRingcdr 18570 LModclmod 18686 LVecclvec 18923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-drng 18572 df-lmod 18688 df-lvec 18924 |
This theorem is referenced by: hlhillvec 36261 |
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