rrx0 - Metamath Proof Explorer

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Theorem rrx0 25459

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 25449	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6871	. 2 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝐻) = (0_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		eqid 2762	. . . . . 6 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
5		eqid 2762	. . . . . 6 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
6		eqid 2762	. . . . . 6 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
7	4, 5, 6	tcphval 25280	. . . . 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 6871	. . 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 6882	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ_fld freeLMod 𝐼)) ∈ V)
11	10	mptexd 7208	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑥(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑥))) ∈ V)
12		eqid 2762	. . . . 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 2762	. . . . 5 ⊢ (0_g‘(ℝ_fld freeLMod 𝐼)) = (0_g‘(ℝ_fld freeLMod 𝐼))
14	12, 13	tng0 24703	. . . 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 21671	. . . . . 6 ⊢ ℝ_fld ∈ Field
18		isfld 20790	. . . . . . 7 ⊢ (ℝ_fld ∈ Field ↔ (ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing))
19		drngring 20786	. . . . . . . 8 ⊢ (ℝ_fld ∈ DivRing → ℝ_fld ∈ Ring)
20	19	adantr 484	. . . . . . 7 ⊢ ((ℝ_fld ∈ DivRing ∧ ℝ_fld ∈ CRing) → ℝ_fld ∈ Ring)
21	18, 20	sylbi 219	. . . . . 6 ⊢ (ℝ_fld ∈ Field → ℝ_fld ∈ Ring)
22	17, 21	ax-mp 5	. . . . 5 ⊢ ℝ_fld ∈ Ring
23		eqid 2762	. . . . . 6 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
24		re0g 21664	. . . . . 6 ⊢ 0 = (0_g‘ℝ_fld)
25	23, 24	frlm0 21806	. . . . 5 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
26	22, 25	mpan 700	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) = (0_g‘(ℝ_fld freeLMod 𝐼)))
27	16, 26	eqtr2id 2810	. . 3 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(ℝ_fld freeLMod 𝐼)) = 0 )
28	9, 15, 27	3eqtr2d 2803	. 2 ⊢ (𝐼 ∈ 𝑉 → (0_g‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = 0 )
29	3, 28	eqtrd 2797	1 ⊢ (𝐼 ∈ 𝑉 → (0_g‘𝐻) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1560 ∈ wcel 2142 Vcvv 3454 {csn 4582 ↦ cmpt 5181 × cxp 5645 ‘cfv 6521 (class class class)co 7396 0cc0 11073 √csqrt 15260 Basecbs 17245 ·_𝑖cip 17291 0_gc0g 17468 Ringcrg 20283 CRingccrg 20284 DivRingcdr 20779 Fieldcfield 20780 ℝ_fldcrefld 21656 freeLMod cfrlm 21798 toNrmGrp ctng 24638 toℂPreHilctcph 25229 ℝ^crrx 25445

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-addf 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-tpos 8206 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9388 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-fz 13513 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-0g 17470 df-prds 17476 df-pws 17478 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-grp 18978 df-minusg 18979 df-sbg 18980 df-subg 19165 df-cmn 19822 df-abl 19823 df-mgp 20187 df-rng 20199 df-ur 20232 df-ring 20285 df-cring 20286 df-oppr 20386 df-dvdsr 20406 df-unit 20407 df-invr 20437 df-dvr 20450 df-subrng 20596 df-subrg 20620 df-drng 20781 df-field 20782 df-lmod 20929 df-lss 20999 df-sra 21240 df-rgmod 21241 df-cnfld 21425 df-refld 21657 df-dsmm 21784 df-frlm 21799 df-tng 24644 df-tcph 25231 df-rrx 25447

This theorem is referenced by: ehl0 25479

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