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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 23990 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
3 | 2 | fveq2d 6674 | . 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 23821 | . . . . 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 6674 | . . 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 6685 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝfld freeLMod 𝐼)) ∈ V) | |
11 | 10 | mptexd 6987 | . . . 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 23252 | . . . 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 20763 | . . . . . 6 ⊢ ℝfld ∈ Field | |
18 | isfld 19511 | . . . . . . 7 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
19 | drngring 19509 | . . . . . . . 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 20756 | . . . . . 6 ⊢ 0 = (0g‘ℝfld) | |
25 | 23, 24 | frlm0 20898 | . . . . 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 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 ↦ cmpt 5146 × cxp 5553 ‘cfv 6355 (class class class)co 7156 0cc0 10537 √csqrt 14592 Basecbs 16483 ·𝑖cip 16570 0gc0g 16713 Ringcrg 19297 CRingccrg 19298 DivRingcdr 19502 Fieldcfield 19503 ℝfldcrefld 20748 freeLMod cfrlm 20890 toNrmGrp ctng 23188 toℂPreHilctcph 23771 ℝ^crrx 23986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-0g 16715 df-prds 16721 df-pws 16723 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-field 19505 df-subrg 19533 df-lmod 19636 df-lss 19704 df-sra 19944 df-rgmod 19945 df-cnfld 20546 df-refld 20749 df-dsmm 20876 df-frlm 20891 df-tng 23194 df-tcph 23773 df-rrx 23988 |
This theorem is referenced by: ehl0 24020 |
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