rrx0 - Metamath Proof Explorer

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Theorem rrx0 25365

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 25355	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6846	. 2 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝐻) = (0_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		eqid 2737	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
5		eqid 2737	. . . . . 6 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
6		eqid 2737	. . . . . 6 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
7	4, 5, 6	tcphval 25186	. . . . 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 6846	. . 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 6857	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ_fld freeLMod 𝐼)) ∈ V)
11	10	mptexd 7180	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))) ∈ V)
12		eqid 2737	. . . . 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 2737	. . . . 5 ⊢ (0_g‘(ℝ_fld freeLMod 𝐼)) = (0_g‘(ℝ_fld freeLMod 𝐼))
14	12, 13	tng0 24599	. . . 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 21586	. . . . . 6 ⊢ ℝ_fld ∈ Field
18		isfld 20685	. . . . . . 7 ⊢ (ℝ_fld ∈ Field ↔ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing))
19		drngring 20681	. . . . . . . 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 2737	. . . . . 6 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
24		re0g 21579	. . . . . 6 ⊢ 0 = (0_g‘ℝ_fld)
25	23, 24	frlm0 21721	. . . . 5 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
26	22, 25	mpan 691	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
27	16, 26	eqtr2id 2785	. . 3 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(ℝ_fld freeLMod 𝐼)) = 0 )
28	9, 15, 27	3eqtr2d 2778	. 2 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = 0 )
29	3, 28	eqtrd 2772	1 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝐻) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3442 {csn 4582 ↦ cmpt 5181 × cxp 5630 ‘cfv 6500 (class class class)co 7368 0cc0 11038 √csqrt 15168 Basecbs 17148 ·_𝑖cip 17194 0_gc0g 17371 Ringcrg 20180 CRingccrg 20181 DivRingcdr 20674 Fieldcfield 20675 ℝ_fldcrefld 21571 freeLMod cfrlm 21713 toNrmGrp ctng 24534 toℂPreHilctcph 25135 ℝ^crrx 25351

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-0g 17373 df-prds 17379 df-pws 17381 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-subrng 20491 df-subrg 20515 df-drng 20676 df-field 20677 df-lmod 20825 df-lss 20895 df-sra 21137 df-rgmod 21138 df-cnfld 21322 df-refld 21572 df-dsmm 21699 df-frlm 21714 df-tng 24540 df-tcph 25137 df-rrx 25353

This theorem is referenced by: ehl0 25385

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