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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 8906 | . . . . 5 ⊢ 1 ∈ ℕ | |
6 | basendx 12486 | . . . . 5 ⊢ (Base‘ndx) = 1 | |
7 | 1lt2 9064 | . . . . 5 ⊢ 1 < 2 | |
8 | 2nn 9056 | . . . . 5 ⊢ 2 ∈ ℕ | |
9 | plusgndx 12536 | . . . . 5 ⊢ (+g‘ndx) = 2 | |
10 | 2lt5 9072 | . . . . 5 ⊢ 2 < 5 | |
11 | 5nn 9059 | . . . . 5 ⊢ 5 ∈ ℕ | |
12 | scandx 12571 | . . . . 5 ⊢ (Scalar‘ndx) = 5 | |
13 | 5, 6, 7, 8, 9, 10, 11, 12 | strle3g 12535 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ + ∈ 𝑋 ∧ 𝐹 ∈ 𝑌) → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} Struct 〈1, 5〉) |
14 | 2, 3, 4, 13 | syl3anc 1238 | . . 3 ⊢ (𝜑 → {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} Struct 〈1, 5〉) |
15 | lmodstr.m | . . . 4 ⊢ (𝜑 → · ∈ 𝑍) | |
16 | 6nn 9060 | . . . . 5 ⊢ 6 ∈ ℕ | |
17 | vscandx 12577 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) = 6 | |
18 | 16, 17 | strle1g 12533 | . . . 4 ⊢ ( · ∈ 𝑍 → {〈( ·𝑠 ‘ndx), · 〉} Struct 〈6, 6〉) |
19 | 15, 18 | syl 14 | . . 3 ⊢ (𝜑 → {〈( ·𝑠 ‘ndx), · 〉} Struct 〈6, 6〉) |
20 | 5lt6 9074 | . . . 4 ⊢ 5 < 6 | |
21 | 20 | a1i 9 | . . 3 ⊢ (𝜑 → 5 < 6) |
22 | 14, 19, 21 | strleund 12531 | . 2 ⊢ (𝜑 → ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉}) Struct 〈1, 6〉) |
23 | 1, 22 | eqbrtrid 4035 | 1 ⊢ (𝜑 → 𝑊 Struct 〈1, 6〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ∪ cun 3127 {csn 3591 {ctp 3593 〈cop 3594 class class class wbr 4000 ‘cfv 5211 1c1 7790 < clt 7969 2c2 8946 5c5 8949 6c6 8950 Struct cstr 12428 ndxcnx 12429 Basecbs 12432 +gcplusg 12505 Scalarcsca 12508 ·𝑠 cvsca 12509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-tp 3599 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-2 8954 df-3 8955 df-4 8956 df-5 8957 df-6 8958 df-n0 9153 df-z 9230 df-uz 9505 df-fz 9983 df-struct 12434 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-sca 12521 df-vsca 12522 |
This theorem is referenced by: lmodbased 12581 lmodplusgd 12582 lmodscad 12583 lmodvscad 12584 |
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