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Mirrors > Home > MPE Home > Th. List > cphcjcl | Structured version Visualization version GIF version |
Description: The scalar field of a subcomplex pre-Hilbert space is closed under conjugation. (Contributed by Mario Carneiro, 11-Oct-2015.) |
Ref | Expression |
---|---|
cphsca.f | ⊢ 𝐹 = (Scalar‘𝑊) |
cphsca.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
cphcjcl | ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾) → (∗‘𝐴) ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphsca.f | . . . . . . 7 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | cphsca.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝐹) | |
3 | 1, 2 | cphsca 23785 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾)) |
4 | 3 | fveq2d 6676 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → (*𝑟‘𝐹) = (*𝑟‘(ℂfld ↾s 𝐾))) |
5 | 2 | fvexi 6686 | . . . . . 6 ⊢ 𝐾 ∈ V |
6 | eqid 2823 | . . . . . . 7 ⊢ (ℂfld ↾s 𝐾) = (ℂfld ↾s 𝐾) | |
7 | cnfldcj 20554 | . . . . . . 7 ⊢ ∗ = (*𝑟‘ℂfld) | |
8 | 6, 7 | ressstarv 16628 | . . . . . 6 ⊢ (𝐾 ∈ V → ∗ = (*𝑟‘(ℂfld ↾s 𝐾))) |
9 | 5, 8 | ax-mp 5 | . . . . 5 ⊢ ∗ = (*𝑟‘(ℂfld ↾s 𝐾)) |
10 | 4, 9 | syl6eqr 2876 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil → (*𝑟‘𝐹) = ∗) |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾) → (*𝑟‘𝐹) = ∗) |
12 | 11 | fveq1d 6674 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾) → ((*𝑟‘𝐹)‘𝐴) = (∗‘𝐴)) |
13 | cphphl 23777 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) | |
14 | 1 | phlsrng 20777 | . . . 4 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝑊 ∈ ℂPreHil → 𝐹 ∈ *-Ring) |
16 | eqid 2823 | . . . 4 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
17 | 16, 2 | srngcl 19628 | . . 3 ⊢ ((𝐹 ∈ *-Ring ∧ 𝐴 ∈ 𝐾) → ((*𝑟‘𝐹)‘𝐴) ∈ 𝐾) |
18 | 15, 17 | sylan 582 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾) → ((*𝑟‘𝐹)‘𝐴) ∈ 𝐾) |
19 | 12, 18 | eqeltrrd 2916 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾) → (∗‘𝐴) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ‘cfv 6357 (class class class)co 7158 ∗ccj 14457 Basecbs 16485 ↾s cress 16486 *𝑟cstv 16569 Scalarcsca 16570 *-Ringcsr 19617 ℂfldccnfld 20547 PreHilcphl 20770 ℂPreHilccph 23772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-cj 14460 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-mhm 17958 df-ghm 18358 df-mgp 19242 df-ur 19254 df-ring 19301 df-rnghom 19469 df-staf 19618 df-srng 19619 df-cnfld 20548 df-phl 20772 df-cph 23774 |
This theorem is referenced by: cphabscl 23791 |
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