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Mirrors > Home > ILE Home > Th. List > ipsmulrd | GIF version |
Description: The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
Ref | Expression |
---|---|
ipspart.a | ⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) |
ipsstrd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
ipsstrd.p | ⊢ (𝜑 → + ∈ 𝑊) |
ipsstrd.r | ⊢ (𝜑 → × ∈ 𝑋) |
ipsstrd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑌) |
ipsstrd.x | ⊢ (𝜑 → · ∈ 𝑄) |
ipsstrd.i | ⊢ (𝜑 → 𝐼 ∈ 𝑍) |
Ref | Expression |
---|---|
ipsmulrd | ⊢ (𝜑 → × = (.r‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulrslid 12566 | . 2 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ) | |
2 | ipspart.a | . . 3 ⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) | |
3 | ipsstrd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
4 | ipsstrd.p | . . 3 ⊢ (𝜑 → + ∈ 𝑊) | |
5 | ipsstrd.r | . . 3 ⊢ (𝜑 → × ∈ 𝑋) | |
6 | ipsstrd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑌) | |
7 | ipsstrd.x | . . 3 ⊢ (𝜑 → · ∈ 𝑄) | |
8 | ipsstrd.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑍) | |
9 | 2, 3, 4, 5, 6, 7, 8 | ipsstrd 12598 | . 2 ⊢ (𝜑 → 𝐴 Struct 〈1, 8〉) |
10 | 1 | simpri 113 | . . . . 5 ⊢ (.r‘ndx) ∈ ℕ |
11 | opexg 4225 | . . . . 5 ⊢ (((.r‘ndx) ∈ ℕ ∧ × ∈ 𝑋) → 〈(.r‘ndx), × 〉 ∈ V) | |
12 | 10, 5, 11 | sylancr 414 | . . . 4 ⊢ (𝜑 → 〈(.r‘ndx), × 〉 ∈ V) |
13 | tpid3g 3706 | . . . 4 ⊢ (〈(.r‘ndx), × 〉 ∈ V → 〈(.r‘ndx), × 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉}) | |
14 | elun1 3302 | . . . 4 ⊢ (〈(.r‘ndx), × 〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} → 〈(.r‘ndx), × 〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉})) | |
15 | 12, 13, 14 | 3syl 17 | . . 3 ⊢ (𝜑 → 〈(.r‘ndx), × 〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉})) |
16 | 15, 2 | eleqtrrdi 2271 | . 2 ⊢ (𝜑 → 〈(.r‘ndx), × 〉 ∈ 𝐴) |
17 | 1, 9, 5, 16 | opelstrsl 12549 | 1 ⊢ (𝜑 → × = (.r‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∪ cun 3127 {ctp 3593 〈cop 3594 ‘cfv 5212 1c1 7800 ℕcn 8905 8c8 8962 ndxcnx 12439 Slot cslot 12441 Basecbs 12442 +gcplusg 12515 .rcmulr 12516 Scalarcsca 12518 ·𝑠 cvsca 12519 ·𝑖cip 12520 |
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 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-distr 7903 ax-i2m1 7904 ax-0lt1 7905 ax-0id 7907 ax-rnegex 7908 ax-cnre 7910 ax-pre-ltirr 7911 ax-pre-ltwlin 7912 ax-pre-lttrn 7913 ax-pre-apti 7914 ax-pre-ltadd 7915 |
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 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-sub 8117 df-neg 8118 df-inn 8906 df-2 8964 df-3 8965 df-4 8966 df-5 8967 df-6 8968 df-7 8969 df-8 8970 df-n0 9163 df-z 9240 df-uz 9515 df-fz 9993 df-struct 12444 df-ndx 12445 df-slot 12446 df-base 12448 df-plusg 12528 df-mulr 12529 df-sca 12531 df-vsca 12532 df-ip 12533 |
This theorem is referenced by: (None) |
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