rrx0 - Metamath Proof Explorer

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Theorem rrx0 25324

Description: The zero ("origin") in a generalized real Euclidean space. (Contributed by AV, 11-Feb-2023.)

**Hypotheses**
Ref	Expression
rrxsca.r	⊢ 𝐻 = (ℝ^‘𝐼)
rrx0.0	⊢ 0 = (𝐼 × {0})

**Assertion**
Ref	Expression
rrx0	⊢ (𝐼 ∈ 𝑉 → (0_g‘𝐻) = 0 )

Proof of Theorem rrx0 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxsca.r	. . . 4 ⊢ 𝐻 = (ℝ^‘𝐼)
2	1	rrxval 25314	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6901	. 2 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝐻) = (0_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		eqid 2728	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
5		eqid 2728	. . . . . 6 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
6		eqid 2728	. . . . . 6 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
7	4, 5, 6	tcphval 25145	. . . . 5 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = ((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))))
8	7	a1i 11	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = ((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥)))))
9	8	fveq2d 6901	. . 3 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (0_g‘((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))))))
10		fvexd 6912	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ_fld freeLMod 𝐼)) ∈ V)
11	10	mptexd 7236	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))) ∈ V)
12		eqid 2728	. . . . 5 ⊢ ((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥)))) = ((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))))
13		eqid 2728	. . . . 5 ⊢ (0_g‘(ℝ_fld freeLMod 𝐼)) = (0_g‘(ℝ_fld freeLMod 𝐼))
14	12, 13	tng0 24554	. . . 4 ⊢ ((𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))) ∈ V → (0_g‘(ℝ_fld freeLMod 𝐼)) = (0_g‘((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))))))
15	11, 14	syl 17	. . 3 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(ℝ_fld freeLMod 𝐼)) = (0_g‘((ℝ_fld freeLMod 𝐼) toNrmGrp (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))))))
16		rrx0.0	. . . 4 ⊢ 0 = (𝐼 × {0})
17		refld 21550	. . . . . 6 ⊢ ℝ_fld ∈ Field
18		isfld 20634	. . . . . . 7 ⊢ (ℝ_fld ∈ Field ↔ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing))
19		drngring 20630	. . . . . . . 8 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
20	19	adantr 480	. . . . . . 7 ⊢ ((ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing) → ℝ_fld ∈ Ring)
21	18, 20	sylbi 216	. . . . . 6 ⊢ (ℝ_fld ∈ Field → ℝ_fld ∈ Ring)
22	17, 21	ax-mp 5	. . . . 5 ⊢ ℝ_fld ∈ Ring
23		eqid 2728	. . . . . 6 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
24		re0g 21543	. . . . . 6 ⊢ 0 = (0_g‘ℝ_fld)
25	23, 24	frlm0 21687	. . . . 5 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
26	22, 25	mpan 689	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
27	16, 26	eqtr2id 2781	. . 3 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(ℝ_fld freeLMod 𝐼)) = 0 )
28	9, 15, 27	3eqtr2d 2774	. 2 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = 0 )
29	3, 28	eqtrd 2768	1 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝐻) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 {csn 4629 ↦ cmpt 5231 × cxp 5676 ‘cfv 6548 (class class class)co 7420 0cc0 11138 √csqrt 15212 Basecbs 17179 ·_𝑖cip 17237 0_gc0g 17420 Ringcrg 20172 CRingccrg 20173 DivRingcdr 20623 Fieldcfield 20624 ℝ_fldcrefld 21535 freeLMod cfrlm 21679 toNrmGrp ctng 24486 toℂPreHilctcph 25094 ℝ^crrx 25310

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-addf 11217

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-0g 17422 df-prds 17428 df-pws 17430 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-subrng 20482 df-subrg 20507 df-drng 20625 df-field 20626 df-lmod 20744 df-lss 20815 df-sra 21057 df-rgmod 21058 df-cnfld 21279 df-refld 21536 df-dsmm 21665 df-frlm 21680 df-tng 24492 df-tcph 25096 df-rrx 25312

This theorem is referenced by: ehl0 25344

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