rrx0 - Metamath Proof Explorer

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Theorem rrx0 25351

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 25341	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6836	. 2 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝐻) = (0_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		eqid 2734	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
5		eqid 2734	. . . . . 6 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
6		eqid 2734	. . . . . 6 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
7	4, 5, 6	tcphval 25172	. . . . 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 6836	. . 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 6847	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ_fld freeLMod 𝐼)) ∈ V)
11	10	mptexd 7168	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))) ∈ V)
12		eqid 2734	. . . . 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 2734	. . . . 5 ⊢ (0_g‘(ℝ_fld freeLMod 𝐼)) = (0_g‘(ℝ_fld freeLMod 𝐼))
14	12, 13	tng0 24585	. . . 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 21572	. . . . . 6 ⊢ ℝ_fld ∈ Field
18		isfld 20671	. . . . . . 7 ⊢ (ℝ_fld ∈ Field ↔ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing))
19		drngring 20667	. . . . . . . 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 217	. . . . . 6 ⊢ (ℝ_fld ∈ Field → ℝ_fld ∈ Ring)
22	17, 21	ax-mp 5	. . . . 5 ⊢ ℝ_fld ∈ Ring
23		eqid 2734	. . . . . 6 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
24		re0g 21565	. . . . . 6 ⊢ 0 = (0_g‘ℝ_fld)
25	23, 24	frlm0 21707	. . . . 5 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
26	22, 25	mpan 690	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
27	16, 26	eqtr2id 2782	. . 3 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(ℝ_fld freeLMod 𝐼)) = 0 )
28	9, 15, 27	3eqtr2d 2775	. 2 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = 0 )
29	3, 28	eqtrd 2769	1 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝐻) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3438 {csn 4578 ↦ cmpt 5177 × cxp 5620 ‘cfv 6490 (class class class)co 7356 0cc0 11024 √csqrt 15154 Basecbs 17134 ·_𝑖cip 17180 0_gc0g 17357 Ringcrg 20166 CRingccrg 20167 DivRingcdr 20660 Fieldcfield 20661 ℝ_fldcrefld 21557 freeLMod cfrlm 21699 toNrmGrp ctng 24520 toℂPreHilctcph 25121 ℝ^crrx 25337

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-addf 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-fz 13422 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-0g 17359 df-prds 17365 df-pws 17367 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19051 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-oppr 20271 df-dvdsr 20291 df-unit 20292 df-invr 20322 df-dvr 20335 df-subrng 20477 df-subrg 20501 df-drng 20662 df-field 20663 df-lmod 20811 df-lss 20881 df-sra 21123 df-rgmod 21124 df-cnfld 21308 df-refld 21558 df-dsmm 21685 df-frlm 21700 df-tng 24526 df-tcph 25123 df-rrx 25339

This theorem is referenced by: ehl0 25371

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