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Theorem rrxbase 24887

Description: The base of the generalized real Euclidean space is the set of functions with finite support. (Contributed by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 22-Jul-2019.)

**Hypotheses**
Ref	Expression
rrxval.r	⊢ 𝐻 = (ℝ^‘𝐼)
rrxbase.b	⊢ 𝐵 = (Base‘𝐻)

**Assertion**
Ref	Expression
rrxbase	⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0})

Distinct variable groups: 𝐵,𝑓 𝑓,𝐼 𝑓,𝑉 Allowed substitution hint: 𝐻(𝑓)

**Proof of Theorem rrxbase**
Step	Hyp	Ref	Expression
1		rrxval.r	. . . . 5 ⊢ 𝐻 = (ℝ^‘𝐼)
2	1	rrxval 24886	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6892	. . 3 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		eqid 2733	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
5		eqid 2733	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
6	4, 5	tcphbas 24718	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
7	3, 6	eqtr4di 2791	. 2 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(ℝ_fld freeLMod 𝐼)))
8		rrxbase.b	. . 3 ⊢ 𝐵 = (Base‘𝐻)
9	8	a1i 11	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘𝐻))
10		refld 21156	. . 3 ⊢ ℝ_fld ∈ Field
11		eqid 2733	. . . 4 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
12		rebase 21143	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
13		re0g 21149	. . . 4 ⊢ 0 = (0_g‘ℝ_fld)
14		eqid 2733	. . . 4 ⊢ {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0}
15	11, 12, 13, 14	frlmbas 21294	. . 3 ⊢ ((ℝ_fld ∈ Field ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
16	10, 15	mpan 689	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
17	7, 9, 16	3eqtr4d 2783	1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0})

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3433 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 ↑_m cmap 8816 finSupp cfsupp 9357 ℝcr 11105 0cc0 11106 Basecbs 17140 Fieldcfield 20305 ℝ_fldcrefld 21141 freeLMod cfrlm 21285 toℂPreHilctcph 24666 ℝ^crrx 24882

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-prds 17389 df-pws 17391 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-subg 18997 df-cmn 19643 df-mgp 19980 df-ur 19997 df-ring 20049 df-cring 20050 df-oppr 20139 df-dvdsr 20160 df-unit 20161 df-invr 20191 df-dvr 20204 df-drng 20306 df-field 20307 df-subrg 20349 df-sra 20773 df-rgmod 20774 df-cnfld 20930 df-refld 21142 df-dsmm 21271 df-frlm 21286 df-tng 24075 df-tcph 24668 df-rrx 24884

This theorem is referenced by: rrxnm 24890 rrxds 24892 rrxmval 24904 rrxmfval 24905 rrxbasefi 24909 rrxmetfi 24911 ehlbase 24914 k0004ss2 42836 rrnprjdstle 44952

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