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Mirrors > Home > ILE Home > Th. List > ipsbased | GIF version |
Description: The base set 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 |
---|---|
ipsbased | ⊢ (𝜑 → 𝐵 = (Base‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipspart.a | . . 3 ⊢ 𝐴 = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉}) | |
2 | ipsstrd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
3 | ipsstrd.p | . . 3 ⊢ (𝜑 → + ∈ 𝑊) | |
4 | ipsstrd.r | . . 3 ⊢ (𝜑 → × ∈ 𝑋) | |
5 | ipsstrd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑌) | |
6 | ipsstrd.x | . . 3 ⊢ (𝜑 → · ∈ 𝑄) | |
7 | ipsstrd.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑍) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ipsstrd 12649 | . 2 ⊢ (𝜑 → 𝐴 Struct 〈1, 8〉) |
9 | basendxnn 12532 | . . . . 5 ⊢ (Base‘ndx) ∈ ℕ | |
10 | opexg 4240 | . . . . 5 ⊢ (((Base‘ndx) ∈ ℕ ∧ 𝐵 ∈ 𝑉) → 〈(Base‘ndx), 𝐵〉 ∈ V) | |
11 | 9, 2, 10 | sylancr 414 | . . . 4 ⊢ (𝜑 → 〈(Base‘ndx), 𝐵〉 ∈ V) |
12 | tpid1g 3716 | . . . 4 ⊢ (〈(Base‘ndx), 𝐵〉 ∈ V → 〈(Base‘ndx), 𝐵〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉}) | |
13 | elun1 3314 | . . . 4 ⊢ (〈(Base‘ndx), 𝐵〉 ∈ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} → 〈(Base‘ndx), 𝐵〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉})) | |
14 | 11, 12, 13 | 3syl 17 | . . 3 ⊢ (𝜑 → 〈(Base‘ndx), 𝐵〉 ∈ ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉, 〈(·𝑖‘ndx), 𝐼〉})) |
15 | 14, 1 | eleqtrrdi 2281 | . 2 ⊢ (𝜑 → 〈(Base‘ndx), 𝐵〉 ∈ 𝐴) |
16 | 8, 2, 15 | opelstrbas 12589 | 1 ⊢ (𝜑 → 𝐵 = (Base‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 Vcvv 2749 ∪ cun 3139 {ctp 3606 〈cop 3607 ‘cfv 5228 1c1 7826 ℕcn 8933 8c8 8990 ndxcnx 12473 Basecbs 12476 +gcplusg 12551 .rcmulr 12552 Scalarcsca 12554 ·𝑠 cvsca 12555 ·𝑖cip 12556 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-tp 3612 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-5 8995 df-6 8996 df-7 8997 df-8 8998 df-n0 9191 df-z 9268 df-uz 9543 df-fz 10023 df-struct 12478 df-ndx 12479 df-slot 12480 df-base 12482 df-plusg 12564 df-mulr 12565 df-sca 12567 df-vsca 12568 df-ip 12569 |
This theorem is referenced by: (None) |
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