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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 19941 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 19698 | . . 3 ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod)) |
12 | 5, 6, 8, 9 | drngpropd 19531 | . . 3 ⊢ (𝜑 → (𝐹 ∈ DivRing ↔ 𝐺 ∈ DivRing)) |
13 | 11, 12 | anbi12d 632 | . 2 ⊢ (𝜑 → ((𝐾 ∈ LMod ∧ 𝐹 ∈ DivRing) ↔ (𝐿 ∈ LMod ∧ 𝐺 ∈ DivRing))) |
14 | 3 | islvec 19878 | . 2 ⊢ (𝐾 ∈ LVec ↔ (𝐾 ∈ LMod ∧ 𝐹 ∈ DivRing)) |
15 | 4 | islvec 19878 | . 2 ⊢ (𝐿 ∈ LVec ↔ (𝐿 ∈ LMod ∧ 𝐺 ∈ DivRing)) |
16 | 13, 14, 15 | 3bitr4g 316 | 1 ⊢ (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 Scalarcsca 16570 ·𝑠 cvsca 16571 DivRingcdr 19504 LModclmod 19636 LVecclvec 19876 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-mulr 16581 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-drng 19506 df-lmod 19638 df-lvec 19877 |
This theorem is referenced by: hlhillvec 39089 |
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