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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 23984 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
3 | 2 | fveq2d 6668 | . 2 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝐻) = (0g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
4 | eqid 2821 | . . . . . 6 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
5 | eqid 2821 | . . . . . 6 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
6 | eqid 2821 | . . . . . 6 ⊢ (·𝑖‘(ℝfld freeLMod 𝐼)) = (·𝑖‘(ℝfld freeLMod 𝐼)) | |
7 | 4, 5, 6 | tcphval 23815 | . . . . 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 6668 | . . 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 6679 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝfld freeLMod 𝐼)) ∈ V) | |
11 | 10 | mptexd 6981 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥))) ∈ V) |
12 | eqid 2821 | . . . . 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 2821 | . . . . 5 ⊢ (0g‘(ℝfld freeLMod 𝐼)) = (0g‘(ℝfld freeLMod 𝐼)) | |
14 | 12, 13 | tng0 23246 | . . . 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 20757 | . . . . . 6 ⊢ ℝfld ∈ Field | |
18 | isfld 19505 | . . . . . . 7 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
19 | drngring 19503 | . . . . . . . 8 ⊢ (ℝfld ∈ DivRing → ℝfld ∈ Ring) | |
20 | 19 | adantr 483 | . . . . . . 7 ⊢ ((ℝfld ∈ DivRing ∧ ℝfld ∈ CRing) → ℝfld ∈ Ring) |
21 | 18, 20 | sylbi 219 | . . . . . 6 ⊢ (ℝfld ∈ Field → ℝfld ∈ Ring) |
22 | 17, 21 | ax-mp 5 | . . . . 5 ⊢ ℝfld ∈ Ring |
23 | eqid 2821 | . . . . . 6 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
24 | re0g 20750 | . . . . . 6 ⊢ 0 = (0g‘ℝfld) | |
25 | 23, 24 | frlm0 20892 | . . . . 5 ⊢ ((ℝfld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {0}) = (0g‘(ℝfld freeLMod 𝐼))) |
26 | 22, 25 | mpan 688 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) = (0g‘(ℝfld freeLMod 𝐼))) |
27 | 16, 26 | syl5req 2869 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (0g‘(ℝfld freeLMod 𝐼)) = 0 ) |
28 | 9, 15, 27 | 3eqtr2d 2862 | . 2 ⊢ (𝐼 ∈ 𝑉 → (0g‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) = 0 ) |
29 | 3, 28 | eqtrd 2856 | 1 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝐻) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {csn 4560 ↦ cmpt 5138 × cxp 5547 ‘cfv 6349 (class class class)co 7150 0cc0 10531 √csqrt 14586 Basecbs 16477 ·𝑖cip 16564 0gc0g 16707 Ringcrg 19291 CRingccrg 19292 DivRingcdr 19496 Fieldcfield 19497 ℝfldcrefld 20742 freeLMod cfrlm 20884 toNrmGrp ctng 23182 toℂPreHilctcph 23765 ℝ^crrx 23980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-0g 16709 df-prds 16715 df-pws 16717 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-field 19499 df-subrg 19527 df-lmod 19630 df-lss 19698 df-sra 19938 df-rgmod 19939 df-cnfld 20540 df-refld 20743 df-dsmm 20870 df-frlm 20885 df-tng 23188 df-tcph 23767 df-rrx 23982 |
This theorem is referenced by: ehl0 24014 |
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