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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 25435 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼))) |
3 | 2 | fveq2d 6911 | . 2 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝐻) = (0g‘(toℂPreHil‘(ℝfld freeLMod 𝐼)))) |
4 | eqid 2735 | . . . . . 6 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)) | |
5 | eqid 2735 | . . . . . 6 ⊢ (Base‘(ℝfld freeLMod 𝐼)) = (Base‘(ℝfld freeLMod 𝐼)) | |
6 | eqid 2735 | . . . . . 6 ⊢ (·𝑖‘(ℝfld freeLMod 𝐼)) = (·𝑖‘(ℝfld freeLMod 𝐼)) | |
7 | 4, 5, 6 | tcphval 25266 | . . . . 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 6911 | . . 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 6922 | . . . . 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 2735 | . . . . 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 2735 | . . . . 5 ⊢ (0g‘(ℝfld freeLMod 𝐼)) = (0g‘(ℝfld freeLMod 𝐼)) | |
14 | 12, 13 | tng0 24675 | . . . 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 21655 | . . . . . 6 ⊢ ℝfld ∈ Field | |
18 | isfld 20757 | . . . . . . 7 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
19 | drngring 20753 | . . . . . . . 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 2735 | . . . . . 6 ⊢ (ℝfld freeLMod 𝐼) = (ℝfld freeLMod 𝐼) | |
24 | re0g 21648 | . . . . . 6 ⊢ 0 = (0g‘ℝfld) | |
25 | 23, 24 | frlm0 21792 | . . . . 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 2788 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (0g‘(ℝfld freeLMod 𝐼)) = 0 ) |
28 | 9, 15, 27 | 3eqtr2d 2781 | . 2 ⊢ (𝐼 ∈ 𝑉 → (0g‘(toℂPreHil‘(ℝfld freeLMod 𝐼))) = 0 ) |
29 | 3, 28 | eqtrd 2775 | 1 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝐻) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 ↦ cmpt 5231 × cxp 5687 ‘cfv 6563 (class class class)co 7431 0cc0 11153 √csqrt 15269 Basecbs 17245 ·𝑖cip 17303 0gc0g 17486 Ringcrg 20251 CRingccrg 20252 DivRingcdr 20746 Fieldcfield 20747 ℝfldcrefld 21640 freeLMod cfrlm 21784 toNrmGrp ctng 24607 toℂPreHilctcph 25215 ℝ^crrx 25431 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-refld 21641 df-dsmm 21770 df-frlm 21785 df-tng 24613 df-tcph 25217 df-rrx 25433 |
This theorem is referenced by: ehl0 25465 |
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