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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 25421 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) | 
| 3 | 2 | fveq2d 6910 | . 2 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝐻) = (0g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) | 
| 4 | eqid 2737 | . . . . . 6 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
| 5 | eqid 2737 | . . . . . 6 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
| 6 | eqid 2737 | . . . . . 6 ⊢ (·𝑖‘(ℝfld freeLMod 𝐼)) = (·𝑖‘(ℝfld freeLMod 𝐼)) | |
| 7 | 4, 5, 6 | tcphval 25252 | . . . . 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 6910 | . . 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 6921 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝfld freeLMod 𝐼)) ∈ V) | |
| 11 | 10 | mptexd 7244 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ (Base‘(ℝfld freeLMod 𝐼)) ↦ (√‘(𝑥(·𝑖‘(ℝfld freeLMod 𝐼))𝑥))) ∈ V) | 
| 12 | eqid 2737 | . . . . 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 2737 | . . . . 5 ⊢ (0g‘(ℝfld freeLMod 𝐼)) = (0g‘(ℝfld freeLMod 𝐼)) | |
| 14 | 12, 13 | tng0 24659 | . . . 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 21637 | . . . . . 6 ⊢ ℝfld ∈ Field | |
| 18 | isfld 20740 | . . . . . . 7 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
| 19 | drngring 20736 | . . . . . . . 8 ⊢ (ℝfld ∈ DivRing → ℝfld ∈ Ring) | |
| 20 | 19 | adantr 480 | . . . . . . 7 ⊢ ((ℝfld ∈ DivRing ∧ ℝfld ∈ CRing) → ℝfld ∈ Ring) | 
| 21 | 18, 20 | sylbi 217 | . . . . . 6 ⊢ (ℝfld ∈ Field → ℝfld ∈ Ring) | 
| 22 | 17, 21 | ax-mp 5 | . . . . 5 ⊢ ℝfld ∈ Ring | 
| 23 | eqid 2737 | . . . . . 6 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
| 24 | re0g 21630 | . . . . . 6 ⊢ 0 = (0g‘ℝfld) | |
| 25 | 23, 24 | frlm0 21774 | . . . . 5 ⊢ ((ℝfld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {0}) = (0g‘(ℝfld freeLMod 𝐼))) | 
| 26 | 22, 25 | mpan 690 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) = (0g‘(ℝfld freeLMod 𝐼))) | 
| 27 | 16, 26 | eqtr2id 2790 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (0g‘(ℝfld freeLMod 𝐼)) = 0 ) | 
| 28 | 9, 15, 27 | 3eqtr2d 2783 | . 2 ⊢ (𝐼 ∈ 𝑉 → (0g‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) = 0 ) | 
| 29 | 3, 28 | eqtrd 2777 | 1 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝐻) = 0 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 ↦ cmpt 5225 × cxp 5683 ‘cfv 6561 (class class class)co 7431 0cc0 11155 √csqrt 15272 Basecbs 17247 ·𝑖cip 17302 0gc0g 17484 Ringcrg 20230 CRingccrg 20231 DivRingcdr 20729 Fieldcfield 20730 ℝfldcrefld 21622 freeLMod cfrlm 21766 toNrmGrp ctng 24591 toℂPreHilctcph 25201 ℝ^crrx 25417 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-subrng 20546 df-subrg 20570 df-drng 20731 df-field 20732 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-refld 21623 df-dsmm 21752 df-frlm 21767 df-tng 24597 df-tcph 25203 df-rrx 25419 | 
| This theorem is referenced by: ehl0 25451 | 
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