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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 | ⊢ 𝐻 = (ℝ^‘𝐼) |
rrx0.0 | ⊢ 0 = (𝐼 × {0}) |
Ref | Expression |
---|---|
rrx0 | ⊢ (𝐼 ∈ 𝑉 → (0g‘𝐻) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrxsca.r | . . . 4 ⊢ 𝐻 = (ℝ^‘𝐼) | |
2 | 1 | rrxval 23562 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
3 | 2 | fveq2d 6441 | . 2 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝐻) = (0g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
4 | eqid 2825 | . . . . . 6 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
5 | eqid 2825 | . . . . . 6 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
6 | eqid 2825 | . . . . . 6 ⊢ (·𝑖‘(ℝfld freeLMod 𝐼)) = (·𝑖‘(ℝfld freeLMod 𝐼)) | |
7 | 4, 5, 6 | tcphval 23393 | . . . . 5 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = ((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))) |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (toℂPreHil‘(ℝfld freeLMod 𝐼)) = ((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥))))) |
9 | 8 | fveq2d 6441 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (0g‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) = (0g‘((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))))) |
10 | fvexd 6452 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝfld freeLMod 𝐼)) ∈ V) | |
11 | 10 | mptexd 6748 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥))) ∈ V) |
12 | eqid 2825 | . . . . 5 ⊢ ((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))) = ((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))) | |
13 | eqid 2825 | . . . . 5 ⊢ (0g‘(ℝfld freeLMod 𝐼)) = (0g‘(ℝfld freeLMod 𝐼)) | |
14 | 12, 13 | tng0 22824 | . . . 4 ⊢ ((𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥))) ∈ V → (0g‘(ℝfld freeLMod 𝐼)) = (0g‘((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))))) |
15 | 11, 14 | syl 17 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (0g‘(ℝfld freeLMod 𝐼)) = (0g‘((ℝfld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥)))))) |
16 | rrx0.0 | . . . 4 ⊢ 0 = (𝐼 × {0}) | |
17 | refld 20333 | . . . . . 6 ⊢ ℝfld ∈ Field | |
18 | isfld 19119 | . . . . . . 7 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
19 | drngring 19117 | . . . . . . . 8 ⊢ (ℝfld ∈ DivRing → ℝfld ∈ Ring) | |
20 | 19 | adantr 474 | . . . . . . 7 ⊢ ((ℝfld ∈ DivRing ∧ ℝfld ∈ CRing) → ℝfld ∈ Ring) |
21 | 18, 20 | sylbi 209 | . . . . . 6 ⊢ (ℝfld ∈ Field → ℝfld ∈ Ring) |
22 | 17, 21 | ax-mp 5 | . . . . 5 ⊢ ℝfld ∈ Ring |
23 | eqid 2825 | . . . . . 6 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
24 | re0g 20326 | . . . . . 6 ⊢ 0 = (0g‘ℝfld) | |
25 | 23, 24 | frlm0 20468 | . . . . 5 ⊢ ((ℝfld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {0}) = (0g‘(ℝfld freeLMod 𝐼))) |
26 | 22, 25 | mpan 681 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) = (0g‘(ℝfld freeLMod 𝐼))) |
27 | 16, 26 | syl5req 2874 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (0g‘(ℝfld freeLMod 𝐼)) = 0 ) |
28 | 9, 15, 27 | 3eqtr2d 2867 | . 2 ⊢ (𝐼 ∈ 𝑉 → (0g‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) = 0 ) |
29 | 3, 28 | eqtrd 2861 | 1 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝐻) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 {csn 4399 ↦ cmpt 4954 × cxp 5344 ‘cfv 6127 (class class class)co 6910 0cc0 10259 √csqrt 14357 Basecbs 16229 ·𝑖cip 16317 0gc0g 16460 Ringcrg 18908 CRingccrg 18909 DivRingcdr 19110 Fieldcfield 19111 ℝfldcrefld 20318 freeLMod cfrlm 20460 toNrmGrp ctng 22760 toℂPreHilctcph 23343 ℝ^crrx 23558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-hom 16336 df-cco 16337 df-0g 16462 df-prds 16468 df-pws 16470 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-cmn 18555 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-drng 19112 df-field 19113 df-subrg 19141 df-lmod 19228 df-lss 19296 df-sra 19540 df-rgmod 19541 df-cnfld 20114 df-refld 20319 df-dsmm 20446 df-frlm 20461 df-tng 22766 df-tcph 23345 df-rrx 23560 |
This theorem is referenced by: ehl0 23592 |
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