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Mirrors > Home > ILE Home > Th. List > lmodplusgd | GIF version |
Description: The additive operation 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 |
---|---|
lmodplusgd | ⊢ (𝜑 → + = (+g‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plusgslid 12624 | . 2 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘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 12675 | . 2 ⊢ (𝜑 → 𝑊 Struct 〈1, 6〉) |
8 | 1 | simpri 113 | . . . . 5 ⊢ (+g‘ndx) ∈ ℕ |
9 | opexg 4246 | . . . . 5 ⊢ (((+g‘ndx) ∈ ℕ ∧ + ∈ 𝑋) → 〈(+g‘ndx), + 〉 ∈ V) | |
10 | 8, 4, 9 | sylancr 414 | . . . 4 ⊢ (𝜑 → 〈(+g‘ndx), + 〉 ∈ V) |
11 | tpid2g 3721 | . . . 4 ⊢ (〈(+g‘ndx), + 〉 ∈ V → 〈(+g‘ndx), + 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉}) | |
12 | elun1 3317 | . . . 4 ⊢ (〈(+g‘ndx), + 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} → 〈(+g‘ndx), + 〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉})) | |
13 | 10, 11, 12 | 3syl 17 | . . 3 ⊢ (𝜑 → 〈(+g‘ndx), + 〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(Scalar‘ndx), 𝐹〉} ∪ {〈( ·𝑠 ‘ndx), · 〉})) |
14 | 13, 2 | eleqtrrdi 2283 | . 2 ⊢ (𝜑 → 〈(+g‘ndx), + 〉 ∈ 𝑊) |
15 | 1, 7, 4, 14 | opelstrsl 12626 | 1 ⊢ (𝜑 → + = (+g‘𝑊)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ∪ cun 3142 {csn 3607 {ctp 3609 〈cop 3610 ‘cfv 5235 1c1 7842 ℕcn 8949 6c6 9004 ndxcnx 12509 Slot cslot 12511 Basecbs 12512 +gcplusg 12589 Scalarcsca 12592 ·𝑠 cvsca 12593 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-tp 3615 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-5 9011 df-6 9012 df-n0 9207 df-z 9284 df-uz 9559 df-fz 10039 df-struct 12514 df-ndx 12515 df-slot 12516 df-base 12518 df-plusg 12602 df-sca 12605 df-vsca 12606 |
This theorem is referenced by: (None) |
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