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Mirrors > Home > ILE Home > Th. List > lmodscad | GIF version |
Description: The set of scalars 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 |
---|---|
lmodscad | ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scaslid 12540 | . 2 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ) | |
2 | lvecfn.w | . . 3 ⊢ 𝑊 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) | |
3 | lmodstr.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
4 | lmodstr.g | . . 3 ⊢ (𝜑 → + ∈ 𝑋) | |
5 | lmodstr.s | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑌) | |
6 | lmodstr.m | . . 3 ⊢ (𝜑 → · ∈ 𝑍) | |
7 | 2, 3, 4, 5, 6 | lmodstrd 12544 | . 2 ⊢ (𝜑 → 𝑊 Struct 〈1, 6〉) |
8 | 1 | simpri 112 | . . . . 5 ⊢ (Scalar‘ndx) ∈ ℕ |
9 | opexg 4211 | . . . . 5 ⊢ (((Scalar‘ndx) ∈ ℕ ∧ 𝐹 ∈ 𝑌) → 〈(Scalar‘ndx), 𝐹〉 ∈ V) | |
10 | 8, 5, 9 | sylancr 412 | . . . 4 ⊢ (𝜑 → 〈(Scalar‘ndx), 𝐹〉 ∈ V) |
11 | tpid3g 3696 | . . . 4 ⊢ (〈(Scalar‘ndx), 𝐹〉 ∈ V → 〈(Scalar‘ndx), 𝐹〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉}) | |
12 | elun1 3294 | . . . 4 ⊢ (〈(Scalar‘ndx), 𝐹〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} → 〈(Scalar‘ndx), 𝐹〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉})) | |
13 | 10, 11, 12 | 3syl 17 | . . 3 ⊢ (𝜑 → 〈(Scalar‘ndx), 𝐹〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉})) |
14 | 13, 2 | eleqtrrdi 2264 | . 2 ⊢ (𝜑 → 〈(Scalar‘ndx), 𝐹〉 ∈ 𝑊) |
15 | 1, 7, 5, 14 | opelstrsl 12507 | 1 ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 Vcvv 2730 ∪ cun 3119 {csn 3581 {ctp 3583 〈cop 3584 ‘cfv 5196 1c1 7768 ℕcn 8871 6c6 8926 ndxcnx 12406 Slot cslot 12408 Basecbs 12409 +gcplusg 12473 Scalarcsca 12476 ·𝑠 cvsca 12477 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-tp 3589 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-inn 8872 df-2 8930 df-3 8931 df-4 8932 df-5 8933 df-6 8934 df-n0 9129 df-z 9206 df-uz 9481 df-fz 9959 df-struct 12411 df-ndx 12412 df-slot 12413 df-base 12415 df-plusg 12486 df-sca 12489 df-vsca 12490 |
This theorem is referenced by: (None) |
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