rrxbase - Metamath Proof Explorer

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Theorem rrxbase 25753

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 25752	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6918	. . 3 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		eqid 2763	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
5		eqid 2763	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
6	4, 5	tcphbas 25584	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
7	3, 6	eqtr4di 2816	. 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 21969	. . 3 ⊢ ℝ_fld ∈ Field
11		eqid 2763	. . . 4 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
12		rebase 21956	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
13		re0g 21962	. . . 4 ⊢ 0 = (0_g‘ℝ_fld)
14		eqid 2763	. . . 4 ⊢ {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0}
15	11, 12, 13, 14	frlmbas 22105	. . 3 ⊢ ((ℝ_fld ∈ Field ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
16	10, 15	mpan 703	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
17	7, 9, 16	3eqtr4d 2808	1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0})

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 {crab 3416 class class class wbr 5106 ‘cfv 6564 (class class class)co 7449 ↑_m cmap 8883 finSupp cfsupp 9389 ℝcr 11236 0cc0 11237 Basecbs 17428 Fieldcfield 21023 ℝ_fldcrefld 21954 freeLMod cfrlm 22096 toℂPreHilctcph 25532 ℝ^crrx 25748

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781 ax-cnex 11293 ax-resscn 11294 ax-1cn 11295 ax-icn 11296 ax-addcl 11297 ax-addrcl 11298 ax-mulcl 11299 ax-mulrcl 11300 ax-mulcom 11301 ax-addass 11302 ax-mulass 11303 ax-distr 11304 ax-i2m1 11305 ax-1ne0 11306 ax-1rid 11307 ax-rnegex 11308 ax-rrecex 11309 ax-cnre 11310 ax-pre-lttri 11311 ax-pre-lttrn 11312 ax-pre-ltadd 11313 ax-pre-mulgt0 11314 ax-pre-sup 11315 ax-addf 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6328 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-riota 7406 df-ov 7452 df-oprab 7453 df-mpo 7454 df-om 7910 df-1st 8033 df-2nd 8034 df-supp 8213 df-tpos 8279 df-frecs 8335 df-wrecs 8366 df-recs 8415 df-rdg 8454 df-1o 8512 df-er 8753 df-map 8885 df-ixp 8962 df-en 9010 df-dom 9011 df-sdom 9012 df-fin 9013 df-fsupp 9390 df-sup 9469 df-pnf 11382 df-mnf 11383 df-xr 11384 df-ltxr 11385 df-le 11386 df-sub 11580 df-neg 11581 df-div 12012 df-nn 12374 df-2 12443 df-3 12444 df-4 12445 df-5 12446 df-6 12447 df-7 12448 df-8 12449 df-9 12450 df-n0 12645 df-z 12732 df-dec 12853 df-uz 13004 df-rp 13159 df-fz 13678 df-seq 14183 df-exp 14243 df-cj 15307 df-re 15308 df-im 15309 df-sqrt 15443 df-abs 15444 df-struct 17366 df-sets 17383 df-slot 17401 df-ndx 17413 df-base 17429 df-ress 17450 df-plusg 17482 df-mulr 17483 df-starv 17484 df-sca 17485 df-vsca 17486 df-ip 17487 df-tset 17488 df-ple 17489 df-ds 17491 df-unif 17492 df-hom 17493 df-cco 17494 df-0g 17653 df-prds 17659 df-pws 17661 df-mgm 18857 df-sgrp 18949 df-mnd 18965 df-grp 19188 df-minusg 19189 df-subg 19374 df-cmn 20038 df-abl 20039 df-mgp 20403 df-rng 20417 df-ur 20450 df-ring 20503 df-cring 20504 df-oppr 20609 df-dvdsr 20629 df-unit 20630 df-invr 20660 df-dvr 20673 df-subrng 20840 df-subrg 20864 df-drng 21024 df-field 21025 df-sra 21490 df-rgmod 21491 df-cnfld 21721 df-refld 21955 df-dsmm 22082 df-frlm 22097 df-tng 24947 df-tcph 25534 df-rrx 25750

This theorem is used by: rrxnm 25756 rrxds 25758 rrxmval 25770 rrxmfval 25771 rrxbasefi 25775 rrxmetfi 25777 ehlbase 25780 k0004ss2 45407 rrnprjdstle 47578

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