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Mirrors > Home > ILE Home > Th. List > lmodstrd | GIF version |
Description: A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Jim Kingdon, 5-Feb-2023.) |
Ref | Expression |
---|---|
lvecfn.w | ⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) |
lmodstr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
lmodstr.g | ⊢ (𝜑 → + ∈ 𝑋) |
lmodstr.s | ⊢ (𝜑 → 𝐹 ∈ 𝑌) |
lmodstr.m | ⊢ (𝜑 → · ∈ 𝑍) |
Ref | Expression |
---|---|
lmodstrd | ⊢ (𝜑 → 𝑊 Struct 〈1, 6〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecfn.w | . 2 ⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) | |
2 | lmodstr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | lmodstr.g | . . . 4 ⊢ (𝜑 → + ∈ 𝑋) | |
4 | lmodstr.s | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑌) | |
5 | 1nn 8859 | . . . . 5 ⊢ 1 ∈ ℕ | |
6 | basendx 12391 | . . . . 5 ⊢ (Base‘ndx) = 1 | |
7 | 1lt2 9017 | . . . . 5 ⊢ 1 < 2 | |
8 | 2nn 9009 | . . . . 5 ⊢ 2 ∈ ℕ | |
9 | plusgndx 12430 | . . . . 5 ⊢ (+g‘ndx) = 2 | |
10 | 2lt5 9025 | . . . . 5 ⊢ 2 < 5 | |
11 | 5nn 9012 | . . . . 5 ⊢ 5 ∈ ℕ | |
12 | scandx 12464 | . . . . 5 ⊢ (Scalar‘ndx) = 5 | |
13 | 5, 6, 7, 8, 9, 10, 11, 12 | strle3g 12429 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑋 ∧ 𝐹 ∈ 𝑌) → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} Struct 〈1, 5〉) |
14 | 2, 3, 4, 13 | syl3anc 1227 | . . 3 ⊢ (𝜑 → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} Struct 〈1, 5〉) |
15 | lmodstr.m | . . . 4 ⊢ (𝜑 → · ∈ 𝑍) | |
16 | 6nn 9013 | . . . . 5 ⊢ 6 ∈ ℕ | |
17 | vscandx 12467 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) = 6 | |
18 | 16, 17 | strle1g 12427 | . . . 4 ⊢ ( · ∈ 𝑍 → {〈( ·𝑠 ‘ndx), · 〉} Struct 〈6, 6〉) |
19 | 15, 18 | syl 14 | . . 3 ⊢ (𝜑 → {〈( ·𝑠 ‘ndx), · 〉} Struct 〈6, 6〉) |
20 | 5lt6 9027 | . . . 4 ⊢ 5 < 6 | |
21 | 20 | a1i 9 | . . 3 ⊢ (𝜑 → 5 < 6) |
22 | 14, 19, 21 | strleund 12425 | . 2 ⊢ (𝜑 → ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) Struct 〈1, 6〉) |
23 | 1, 22 | eqbrtrid 4011 | 1 ⊢ (𝜑 → 𝑊 Struct 〈1, 6〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ∪ cun 3109 {csn 3570 {ctp 3572 〈cop 3573 class class class wbr 3976 ‘cfv 5182 1c1 7745 < clt 7924 2c2 8899 5c5 8902 6c6 8903 Struct cstr 12333 ndxcnx 12334 Basecbs 12337 +gcplusg 12399 Scalarcsca 12402 ·𝑠 cvsca 12403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-tp 3578 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 df-struct 12339 df-ndx 12340 df-slot 12341 df-base 12343 df-plusg 12412 df-sca 12415 df-vsca 12416 |
This theorem is referenced by: lmodbased 12471 lmodplusgd 12472 lmodscad 12473 lmodvscad 12474 |
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