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Mirrors > Home > ILE Home > Th. List > lmodbased | GIF version |
Description: The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-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 |
---|---|
lmodbased | ⊢ (𝜑 → 𝐵 = (Base‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lvecfn.w | . . 3 ⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) | |
2 | lmodstr.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | lmodstr.g | . . 3 ⊢ (𝜑 → + ∈ 𝑋) | |
4 | lmodstr.s | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑌) | |
5 | lmodstr.m | . . 3 ⊢ (𝜑 → · ∈ 𝑍) | |
6 | 1, 2, 3, 4, 5 | lmodstrd 12095 | . 2 ⊢ (𝜑 → 𝑊 Struct 〈1, 6〉) |
7 | basendxnn 12017 | . . . . 5 ⊢ (Base‘ndx) ∈ ℕ | |
8 | opexg 4150 | . . . . 5 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ 𝑉) → 〈(Base‘ndx), 𝐵〉 ∈ V) | |
9 | 7, 2, 8 | sylancr 410 | . . . 4 ⊢ (𝜑 → 〈(Base‘ndx), 𝐵〉 ∈ V) |
10 | tpid1g 3635 | . . . 4 ⊢ (〈(Base‘ndx), 𝐵〉 ∈ V → 〈(Base‘ndx), 𝐵〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉}) | |
11 | elun1 3243 | . . . 4 ⊢ (〈(Base‘ndx), 𝐵〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} → 〈(Base‘ndx), 𝐵〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉})) | |
12 | 9, 10, 11 | 3syl 17 | . . 3 ⊢ (𝜑 → 〈(Base‘ndx), 𝐵〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉})) |
13 | 12, 1 | eleqtrrdi 2233 | . 2 ⊢ (𝜑 → 〈(Base‘ndx), 𝐵〉 ∈ 𝑊) |
14 | 6, 2, 13 | opelstrbas 12059 | 1 ⊢ (𝜑 → 𝐵 = (Base‘𝑊)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 {csn 3527 {ctp 3529 〈cop 3530 ‘cfv 5123 1c1 7624 ℕcn 8723 6c6 8778 ndxcnx 11959 Basecbs 11962 +gcplusg 12024 Scalarcsca 12027 ·𝑠 cvsca 12028 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-tp 3535 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-5 8785 df-6 8786 df-n0 8981 df-z 9058 df-uz 9330 df-fz 9794 df-struct 11964 df-ndx 11965 df-slot 11966 df-base 11968 df-plusg 12037 df-sca 12040 df-vsca 12041 |
This theorem is referenced by: (None) |
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