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Mirrors > Home > MPE Home > Th. List > rrxprds | Structured version Visualization version GIF version |
Description: Expand the definition of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
rrxval.r | β’ π» = (β^βπΌ) |
rrxbase.b | β’ π΅ = (Baseβπ») |
Ref | Expression |
---|---|
rrxprds | β’ (πΌ β π β π» = (toβPreHilβ((βfldXs(πΌ Γ {((subringAlg ββfld)ββ)})) βΎs π΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxval.r | . . 3 β’ π» = (β^βπΌ) | |
2 | 1 | rrxval 25335 | . 2 β’ (πΌ β π β π» = (toβPreHilβ(βfld freeLMod πΌ))) |
3 | refld 21558 | . . . . 5 β’ βfld β Field | |
4 | eqid 2728 | . . . . . 6 β’ (βfld freeLMod πΌ) = (βfld freeLMod πΌ) | |
5 | eqid 2728 | . . . . . 6 β’ (Baseβ(βfld freeLMod πΌ)) = (Baseβ(βfld freeLMod πΌ)) | |
6 | 4, 5 | frlmpws 21691 | . . . . 5 β’ ((βfld β Field β§ πΌ β π) β (βfld freeLMod πΌ) = (((ringLModββfld) βs πΌ) βΎs (Baseβ(βfld freeLMod πΌ)))) |
7 | 3, 6 | mpan 688 | . . . 4 β’ (πΌ β π β (βfld freeLMod πΌ) = (((ringLModββfld) βs πΌ) βΎs (Baseβ(βfld freeLMod πΌ)))) |
8 | fvex 6915 | . . . . . . 7 β’ ((subringAlg ββfld)ββ) β V | |
9 | rlmval 21091 | . . . . . . . . . 10 β’ (ringLModββfld) = ((subringAlg ββfld)β(Baseββfld)) | |
10 | rebase 21545 | . . . . . . . . . . 11 β’ β = (Baseββfld) | |
11 | 10 | fveq2i 6905 | . . . . . . . . . 10 β’ ((subringAlg ββfld)ββ) = ((subringAlg ββfld)β(Baseββfld)) |
12 | 9, 11 | eqtr4i 2759 | . . . . . . . . 9 β’ (ringLModββfld) = ((subringAlg ββfld)ββ) |
13 | 12 | oveq1i 7436 | . . . . . . . 8 β’ ((ringLModββfld) βs πΌ) = (((subringAlg ββfld)ββ) βs πΌ) |
14 | 10 | ressid 17232 | . . . . . . . . . 10 β’ (βfld β Field β (βfld βΎs β) = βfld) |
15 | 3, 14 | ax-mp 5 | . . . . . . . . 9 β’ (βfld βΎs β) = βfld |
16 | eqidd 2729 | . . . . . . . . . . 11 β’ (β€ β ((subringAlg ββfld)ββ) = ((subringAlg ββfld)ββ)) | |
17 | 10 | eqimssi 4042 | . . . . . . . . . . . 12 β’ β β (Baseββfld) |
18 | 17 | a1i 11 | . . . . . . . . . . 11 β’ (β€ β β β (Baseββfld)) |
19 | 16, 18 | srasca 21076 | . . . . . . . . . 10 β’ (β€ β (βfld βΎs β) = (Scalarβ((subringAlg ββfld)ββ))) |
20 | 19 | mptru 1540 | . . . . . . . . 9 β’ (βfld βΎs β) = (Scalarβ((subringAlg ββfld)ββ)) |
21 | 15, 20 | eqtr3i 2758 | . . . . . . . 8 β’ βfld = (Scalarβ((subringAlg ββfld)ββ)) |
22 | 13, 21 | pwsval 17475 | . . . . . . 7 β’ ((((subringAlg ββfld)ββ) β V β§ πΌ β π) β ((ringLModββfld) βs πΌ) = (βfldXs(πΌ Γ {((subringAlg ββfld)ββ)}))) |
23 | 8, 22 | mpan 688 | . . . . . 6 β’ (πΌ β π β ((ringLModββfld) βs πΌ) = (βfldXs(πΌ Γ {((subringAlg ββfld)ββ)}))) |
24 | 23 | eqcomd 2734 | . . . . 5 β’ (πΌ β π β (βfldXs(πΌ Γ {((subringAlg ββfld)ββ)})) = ((ringLModββfld) βs πΌ)) |
25 | 2 | fveq2d 6906 | . . . . . 6 β’ (πΌ β π β (Baseβπ») = (Baseβ(toβPreHilβ(βfld freeLMod πΌ)))) |
26 | rrxbase.b | . . . . . 6 β’ π΅ = (Baseβπ») | |
27 | eqid 2728 | . . . . . . 7 β’ (toβPreHilβ(βfld freeLMod πΌ)) = (toβPreHilβ(βfld freeLMod πΌ)) | |
28 | 27, 5 | tcphbas 25167 | . . . . . 6 β’ (Baseβ(βfld freeLMod πΌ)) = (Baseβ(toβPreHilβ(βfld freeLMod πΌ))) |
29 | 25, 26, 28 | 3eqtr4g 2793 | . . . . 5 β’ (πΌ β π β π΅ = (Baseβ(βfld freeLMod πΌ))) |
30 | 24, 29 | oveq12d 7444 | . . . 4 β’ (πΌ β π β ((βfldXs(πΌ Γ {((subringAlg ββfld)ββ)})) βΎs π΅) = (((ringLModββfld) βs πΌ) βΎs (Baseβ(βfld freeLMod πΌ)))) |
31 | 7, 30 | eqtr4d 2771 | . . 3 β’ (πΌ β π β (βfld freeLMod πΌ) = ((βfldXs(πΌ Γ {((subringAlg ββfld)ββ)})) βΎs π΅)) |
32 | 31 | fveq2d 6906 | . 2 β’ (πΌ β π β (toβPreHilβ(βfld freeLMod πΌ)) = (toβPreHilβ((βfldXs(πΌ Γ {((subringAlg ββfld)ββ)})) βΎs π΅))) |
33 | 2, 32 | eqtrd 2768 | 1 β’ (πΌ β π β π» = (toβPreHilβ((βfldXs(πΌ Γ {((subringAlg ββfld)ββ)})) βΎs π΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β€wtru 1534 β wcel 2098 Vcvv 3473 β wss 3949 {csn 4632 Γ cxp 5680 βcfv 6553 (class class class)co 7426 βcr 11145 Basecbs 17187 βΎs cress 17216 Scalarcsca 17243 Xscprds 17434 βs cpws 17435 Fieldcfield 20632 subringAlg csra 21063 ringLModcrglmod 21064 βfldcrefld 21543 freeLMod cfrlm 21687 toβPreHilctcph 25115 β^crrx 25331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-fz 13525 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-0g 17430 df-prds 17436 df-pws 17438 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-subg 19085 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-subrng 20490 df-subrg 20515 df-drng 20633 df-field 20634 df-sra 21065 df-rgmod 21066 df-cnfld 21287 df-refld 21544 df-dsmm 21673 df-frlm 21688 df-tng 24513 df-tcph 25117 df-rrx 25333 |
This theorem is referenced by: rrxip 25338 rrxsca 25344 |
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