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Mirrors > Home > MPE Home > Th. List > lmodstr | Structured version Visualization version GIF version |
Description: A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
lvecfn.w | ⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) |
Ref | Expression |
---|---|
lmodstr | ⊢ 𝑊 Struct 〈1, 6〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecfn.w | . 2 ⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) | |
2 | 1nn 11635 | . . . 4 ⊢ 1 ∈ ℕ | |
3 | basendx 16530 | . . . 4 ⊢ (Base‘ndx) = 1 | |
4 | 1lt2 11795 | . . . 4 ⊢ 1 < 2 | |
5 | 2nn 11697 | . . . 4 ⊢ 2 ∈ ℕ | |
6 | plusgndx 16578 | . . . 4 ⊢ (+g‘ndx) = 2 | |
7 | 2lt5 11803 | . . . 4 ⊢ 2 < 5 | |
8 | 5nn 11710 | . . . 4 ⊢ 5 ∈ ℕ | |
9 | scandx 16615 | . . . 4 ⊢ (Scalar‘ndx) = 5 | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | strle3 16577 | . . 3 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} Struct 〈1, 5〉 |
11 | 6nn 11713 | . . . 4 ⊢ 6 ∈ ℕ | |
12 | vscandx 16617 | . . . 4 ⊢ ( ·𝑠 ‘ndx) = 6 | |
13 | 11, 12 | strle1 16575 | . . 3 ⊢ {〈( ·𝑠 ‘ndx), · 〉} Struct 〈6, 6〉 |
14 | 5lt6 11805 | . . 3 ⊢ 5 < 6 | |
15 | 10, 13, 14 | strleun 16574 | . 2 ⊢ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) Struct 〈1, 6〉 |
16 | 1, 15 | eqbrtri 5073 | 1 ⊢ 𝑊 Struct 〈1, 6〉 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3922 {csn 4553 {ctp 4557 〈cop 4559 class class class wbr 5052 ‘cfv 6341 1c1 10524 2c2 11679 5c5 11682 6c6 11683 Struct cstr 16462 ndxcnx 16463 Basecbs 16466 +gcplusg 16548 Scalarcsca 16551 ·𝑠 cvsca 16552 |
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 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
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-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-n0 11885 df-z 11969 df-uz 12231 df-fz 12883 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-plusg 16561 df-sca 16564 df-vsca 16565 |
This theorem is referenced by: lmodbase 16620 lmodplusg 16621 lmodsca 16622 lmodvsca 16623 phlstr 16636 |
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