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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 25259 | . . 3 β’ (πΌ β π β π» = (toβPreHilβ(βfld freeLMod πΌ))) |
3 | 2 | fveq2d 6886 | . 2 β’ (πΌ β π β (0gβπ») = (0gβ(toβPreHilβ(βfld freeLMod πΌ)))) |
4 | eqid 2724 | . . . . . 6 β’ (toβPreHilβ(βfld freeLMod πΌ)) = (toβPreHilβ(βfld freeLMod πΌ)) | |
5 | eqid 2724 | . . . . . 6 β’ (Baseβ(βfld freeLMod πΌ)) = (Baseβ(βfld freeLMod πΌ)) | |
6 | eqid 2724 | . . . . . 6 β’ (Β·πβ(βfld freeLMod πΌ)) = (Β·πβ(βfld freeLMod πΌ)) | |
7 | 4, 5, 6 | tcphval 25090 | . . . . 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 6886 | . . 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 6897 | . . . . 5 β’ (πΌ β π β (Baseβ(βfld freeLMod πΌ)) β V) | |
11 | 10 | mptexd 7218 | . . . 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 2724 | . . . . 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 2724 | . . . . 5 β’ (0gβ(βfld freeLMod πΌ)) = (0gβ(βfld freeLMod πΌ)) | |
14 | 12, 13 | tng0 24499 | . . . 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 21501 | . . . . . 6 β’ βfld β Field | |
18 | isfld 20594 | . . . . . . 7 β’ (βfld β Field β (βfld β DivRing β§ βfld β CRing)) | |
19 | drngring 20590 | . . . . . . . 8 β’ (βfld β DivRing β βfld β Ring) | |
20 | 19 | adantr 480 | . . . . . . 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 2724 | . . . . . 6 β’ (βfld freeLMod πΌ) = (βfld freeLMod πΌ) | |
24 | re0g 21494 | . . . . . 6 β’ 0 = (0gββfld) | |
25 | 23, 24 | frlm0 21638 | . . . . 5 β’ ((βfld β Ring β§ πΌ β π) β (πΌ Γ {0}) = (0gβ(βfld freeLMod πΌ))) |
26 | 22, 25 | mpan 687 | . . . 4 β’ (πΌ β π β (πΌ Γ {0}) = (0gβ(βfld freeLMod πΌ))) |
27 | 16, 26 | eqtr2id 2777 | . . 3 β’ (πΌ β π β (0gβ(βfld freeLMod πΌ)) = 0 ) |
28 | 9, 15, 27 | 3eqtr2d 2770 | . 2 β’ (πΌ β π β (0gβ(toβPreHilβ(βfld freeLMod πΌ))) = 0 ) |
29 | 3, 28 | eqtrd 2764 | 1 β’ (πΌ β π β (0gβπ») = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3466 {csn 4621 β¦ cmpt 5222 Γ cxp 5665 βcfv 6534 (class class class)co 7402 0cc0 11107 βcsqrt 15182 Basecbs 17149 Β·πcip 17207 0gc0g 17390 Ringcrg 20134 CRingccrg 20135 DivRingcdr 20583 Fieldcfield 20584 βfldcrefld 21486 freeLMod cfrlm 21630 toNrmGrp ctng 24431 toβPreHilctcph 25039 β^crrx 25255 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-0g 17392 df-prds 17398 df-pws 17400 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-subrng 20442 df-subrg 20467 df-drng 20585 df-field 20586 df-lmod 20704 df-lss 20775 df-sra 21017 df-rgmod 21018 df-cnfld 21235 df-refld 21487 df-dsmm 21616 df-frlm 21631 df-tng 24437 df-tcph 25041 df-rrx 25257 |
This theorem is referenced by: ehl0 25289 |
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