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Theorem rrx0 25349

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 25339	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6880	. 2 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝐻) = (0_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		eqid 2735	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
5		eqid 2735	. . . . . 6 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
6		eqid 2735	. . . . . 6 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
7	4, 5, 6	tcphval 25170	. . . . 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 6880	. . 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 6891	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ_fld freeLMod 𝐼)) ∈ V)
11	10	mptexd 7216	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))) ∈ V)
12		eqid 2735	. . . . 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 2735	. . . . 5 ⊢ (0_g‘(ℝ_fld freeLMod 𝐼)) = (0_g‘(ℝ_fld freeLMod 𝐼))
14	12, 13	tng0 24582	. . . 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 21579	. . . . . 6 ⊢ ℝ_fld ∈ Field
18		isfld 20700	. . . . . . 7 ⊢ (ℝ_fld ∈ Field ↔ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing))
19		drngring 20696	. . . . . . . 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 2735	. . . . . 6 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
24		re0g 21572	. . . . . 6 ⊢ 0 = (0_g‘ℝ_fld)
25	23, 24	frlm0 21714	. . . . 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 2783	. . 3 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(ℝ_fld freeLMod 𝐼)) = 0 )
28	9, 15, 27	3eqtr2d 2776	. 2 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = 0 )
29	3, 28	eqtrd 2770	1 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝐻) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3459 {csn 4601 ↦ cmpt 5201 × cxp 5652 ‘cfv 6531 (class class class)co 7405 0cc0 11129 √csqrt 15252 Basecbs 17228 ·_𝑖cip 17276 0_gc0g 17453 Ringcrg 20193 CRingccrg 20194 DivRingcdr 20689 Fieldcfield 20690 ℝ_fldcrefld 21564 freeLMod cfrlm 21706 toNrmGrp ctng 24517 toℂPreHilctcph 25119 ℝ^crrx 25335

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 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-addf 11208

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-fz 13525 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-0g 17455 df-prds 17461 df-pws 17463 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-grp 18919 df-minusg 18920 df-sbg 18921 df-subg 19106 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-subrng 20506 df-subrg 20530 df-drng 20691 df-field 20692 df-lmod 20819 df-lss 20889 df-sra 21131 df-rgmod 21132 df-cnfld 21316 df-refld 21565 df-dsmm 21692 df-frlm 21707 df-tng 24523 df-tcph 25121 df-rrx 25337

This theorem is referenced by: ehl0 25369

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