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Mirrors > Home > MPE Home > Th. List > rrx0 | Structured version Visualization version GIF version |
Description: The zero ("origin") in a generalized real Euclidean space. (Contributed by AV, 11-Feb-2023.) |
Ref | Expression |
---|---|
rrxsca.r | β’ π» = (β^βπΌ) |
rrx0.0 | β’ 0 = (πΌ Γ {0}) |
Ref | Expression |
---|---|
rrx0 | β’ (πΌ β π β (0gβπ») = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxsca.r | . . . 4 β’ π» = (β^βπΌ) | |
2 | 1 | rrxval 24754 | . . 3 β’ (πΌ β π β π» = (toβPreHilβ(βfld freeLMod πΌ))) |
3 | 2 | fveq2d 6847 | . 2 β’ (πΌ β π β (0gβπ») = (0gβ(toβPreHilβ(βfld freeLMod πΌ)))) |
4 | eqid 2737 | . . . . . 6 β’ (toβPreHilβ(βfld freeLMod πΌ)) = (toβPreHilβ(βfld freeLMod πΌ)) | |
5 | eqid 2737 | . . . . . 6 β’ (Baseβ(βfld freeLMod πΌ)) = (Baseβ(βfld freeLMod πΌ)) | |
6 | eqid 2737 | . . . . . 6 β’ (Β·πβ(βfld freeLMod πΌ)) = (Β·πβ(βfld freeLMod πΌ)) | |
7 | 4, 5, 6 | tcphval 24585 | . . . . 5 β’ (toβPreHilβ(βfld freeLMod πΌ)) = ((βfld freeLMod πΌ) toNrmGrp (π₯ β (Baseβ(βfld freeLMod πΌ)) β¦ (ββ(π₯(Β·πβ(βfld freeLMod πΌ))π₯)))) |
8 | 7 | a1i 11 | . . . 4 β’ (πΌ β π β (toβPreHilβ(βfld freeLMod πΌ)) = ((βfld freeLMod πΌ) toNrmGrp (π₯ β (Baseβ(βfld freeLMod πΌ)) β¦ (ββ(π₯(Β·πβ(βfld freeLMod πΌ))π₯))))) |
9 | 8 | fveq2d 6847 | . . 3 β’ (πΌ β π β (0gβ(toβPreHilβ(βfld freeLMod πΌ))) = (0gβ((βfld freeLMod πΌ) toNrmGrp (π₯ β (Baseβ(βfld freeLMod πΌ)) β¦ (ββ(π₯(Β·πβ(βfld freeLMod πΌ))π₯)))))) |
10 | fvexd 6858 | . . . . 5 β’ (πΌ β π β (Baseβ(βfld freeLMod πΌ)) β V) | |
11 | 10 | mptexd 7175 | . . . 4 β’ (πΌ β π β (π₯ β (Baseβ(βfld freeLMod πΌ)) β¦ (ββ(π₯(Β·πβ(βfld freeLMod πΌ))π₯))) β V) |
12 | eqid 2737 | . . . . 5 β’ ((βfld freeLMod πΌ) toNrmGrp (π₯ β (Baseβ(βfld freeLMod πΌ)) β¦ (ββ(π₯(Β·πβ(βfld freeLMod πΌ))π₯)))) = ((βfld freeLMod πΌ) toNrmGrp (π₯ β (Baseβ(βfld freeLMod πΌ)) β¦ (ββ(π₯(Β·πβ(βfld freeLMod πΌ))π₯)))) | |
13 | eqid 2737 | . . . . 5 β’ (0gβ(βfld freeLMod πΌ)) = (0gβ(βfld freeLMod πΌ)) | |
14 | 12, 13 | tng0 24005 | . . . 4 β’ ((π₯ β (Baseβ(βfld freeLMod πΌ)) β¦ (ββ(π₯(Β·πβ(βfld freeLMod πΌ))π₯))) β V β (0gβ(βfld freeLMod πΌ)) = (0gβ((βfld freeLMod πΌ) toNrmGrp (π₯ β (Baseβ(βfld freeLMod πΌ)) β¦ (ββ(π₯(Β·πβ(βfld freeLMod πΌ))π₯)))))) |
15 | 11, 14 | syl 17 | . . 3 β’ (πΌ β π β (0gβ(βfld freeLMod πΌ)) = (0gβ((βfld freeLMod πΌ) toNrmGrp (π₯ β (Baseβ(βfld freeLMod πΌ)) β¦ (ββ(π₯(Β·πβ(βfld freeLMod πΌ))π₯)))))) |
16 | rrx0.0 | . . . 4 β’ 0 = (πΌ Γ {0}) | |
17 | refld 21026 | . . . . . 6 β’ βfld β Field | |
18 | isfld 20197 | . . . . . . 7 β’ (βfld β Field β (βfld β DivRing β§ βfld β CRing)) | |
19 | drngring 20193 | . . . . . . . 8 β’ (βfld β DivRing β βfld β Ring) | |
20 | 19 | adantr 482 | . . . . . . 7 β’ ((βfld β DivRing β§ βfld β CRing) β βfld β Ring) |
21 | 18, 20 | sylbi 216 | . . . . . 6 β’ (βfld β Field β βfld β Ring) |
22 | 17, 21 | ax-mp 5 | . . . . 5 β’ βfld β Ring |
23 | eqid 2737 | . . . . . 6 β’ (βfld freeLMod πΌ) = (βfld freeLMod πΌ) | |
24 | re0g 21019 | . . . . . 6 β’ 0 = (0gββfld) | |
25 | 23, 24 | frlm0 21163 | . . . . 5 β’ ((βfld β Ring β§ πΌ β π) β (πΌ Γ {0}) = (0gβ(βfld freeLMod πΌ))) |
26 | 22, 25 | mpan 689 | . . . 4 β’ (πΌ β π β (πΌ Γ {0}) = (0gβ(βfld freeLMod πΌ))) |
27 | 16, 26 | eqtr2id 2790 | . . 3 β’ (πΌ β π β (0gβ(βfld freeLMod πΌ)) = 0 ) |
28 | 9, 15, 27 | 3eqtr2d 2783 | . 2 β’ (πΌ β π β (0gβ(toβPreHilβ(βfld freeLMod πΌ))) = 0 ) |
29 | 3, 28 | eqtrd 2777 | 1 β’ (πΌ β π β (0gβπ») = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3446 {csn 4587 β¦ cmpt 5189 Γ cxp 5632 βcfv 6497 (class class class)co 7358 0cc0 11052 βcsqrt 15119 Basecbs 17084 Β·πcip 17139 0gc0g 17322 Ringcrg 19965 CRingccrg 19966 DivRingcdr 20186 Fieldcfield 20187 βfldcrefld 21011 freeLMod cfrlm 21155 toNrmGrp ctng 23937 toβPreHilctcph 24534 β^crrx 24750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-addf 11131 ax-mulf 11132 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-7 12222 df-8 12223 df-9 12224 df-n0 12415 df-z 12501 df-dec 12620 df-uz 12765 df-fz 13426 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-ress 17114 df-plusg 17147 df-mulr 17148 df-starv 17149 df-sca 17150 df-vsca 17151 df-ip 17152 df-tset 17153 df-ple 17154 df-ds 17156 df-unif 17157 df-hom 17158 df-cco 17159 df-0g 17324 df-prds 17330 df-pws 17332 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-grp 18752 df-minusg 18753 df-sbg 18754 df-subg 18926 df-cmn 19565 df-mgp 19898 df-ur 19915 df-ring 19967 df-cring 19968 df-oppr 20050 df-dvdsr 20071 df-unit 20072 df-invr 20102 df-dvr 20113 df-drng 20188 df-field 20189 df-subrg 20223 df-lmod 20327 df-lss 20396 df-sra 20636 df-rgmod 20637 df-cnfld 20800 df-refld 21012 df-dsmm 21141 df-frlm 21156 df-tng 23943 df-tcph 24536 df-rrx 24752 |
This theorem is referenced by: ehl0 24784 |
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