rrxbase - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > rrxbase		Structured version Visualization version GIF version

Theorem rrxbase 23985

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 23984	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6669	. . 3 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		eqid 2821	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
5		eqid 2821	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
6	4, 5	tcphbas 23816	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
7	3, 6	syl6eqr 2874	. 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 20757	. . 3 ⊢ ℝ_fld ∈ Field
11		eqid 2821	. . . 4 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
12		rebase 20744	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
13		re0g 20750	. . . 4 ⊢ 0 = (0_g‘ℝ_fld)
14		eqid 2821	. . . 4 ⊢ {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0}
15	11, 12, 13, 14	frlmbas 20893	. . 3 ⊢ ((ℝ_fld ∈ Field ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
16	10, 15	mpan 688	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
17	7, 9, 16	3eqtr4d 2866	1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0})

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ↑_m cmap 8400 finSupp cfsupp 8827 ℝcr 10530 0cc0 10531 Basecbs 16477 Fieldcfield 19497 ℝ_fldcrefld 20742 freeLMod cfrlm 20884 toℂPreHilctcph 23765 ℝ^crrx 23980

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12887 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-0g 16709 df-prds 16715 df-pws 16717 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-subg 18270 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-field 19499 df-subrg 19527 df-sra 19938 df-rgmod 19939 df-cnfld 20540 df-refld 20743 df-dsmm 20870 df-frlm 20885 df-tng 23188 df-tcph 23767 df-rrx 23982

This theorem is referenced by: rrxnm 23988 rrxds 23990 rrxmval 24002 rrxmfval 24003 rrxbasefi 24007 rrxmetfi 24009 ehlbase 24012 k0004ss2 40495 rrnprjdstle 42579

W3C validator