rrxbase - Metamath Proof Explorer

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Theorem rrxbase 25437

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 25436	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
3	2	fveq2d 6865	. . 3 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
4		eqid 2761	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
5		eqid 2761	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
6	4, 5	tcphbas 25268	. . 3 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
7	3, 6	eqtr4di 2814	. 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 21658	. . 3 ⊢ ℝ_fld ∈ Field
11		eqid 2761	. . . 4 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
12		rebase 21645	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
13		re0g 21651	. . . 4 ⊢ 0 = (0_g‘ℝ_fld)
14		eqid 2761	. . . 4 ⊢ {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0}
15	11, 12, 13, 14	frlmbas 21794	. . 3 ⊢ ((ℝ_fld ∈ Field ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
16	10, 15	mpan 700	. 2 ⊢ (𝐼 ∈ 𝑉 → {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} = (Base‘(ℝ_fld freeLMod 𝐼)))
17	7, 9, 16	3eqtr4d 2806	1 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0})

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 {crab 3413 class class class wbr 5097 ‘cfv 6515 (class class class)co 7390 ↑_m cmap 8801 finSupp cfsupp 9300 ℝcr 11065 0cc0 11066 Basecbs 17235 Fieldcfield 20766 ℝ_fldcrefld 21643 freeLMod cfrlm 21785 toℂPreHilctcph 25216 ℝ^crrx 25432

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-pre-sup 11144 ax-addf 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-tpos 8199 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-er 8671 df-map 8803 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9301 df-sup 9381 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-z 12562 df-dec 12682 df-uz 12833 df-rp 12987 df-fz 13506 df-seq 14008 df-exp 14068 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-0g 17460 df-prds 17466 df-pws 17468 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-grp 18968 df-minusg 18969 df-subg 19155 df-cmn 19812 df-abl 19813 df-mgp 20177 df-rng 20189 df-ur 20218 df-ring 20271 df-cring 20272 df-oppr 20372 df-dvdsr 20392 df-unit 20393 df-invr 20423 df-dvr 20436 df-subrng 20582 df-subrg 20606 df-drng 20767 df-field 20768 df-sra 21227 df-rgmod 21228 df-cnfld 21412 df-refld 21644 df-dsmm 21771 df-frlm 21786 df-tng 24631 df-tcph 25218 df-rrx 25434

This theorem is referenced by: rrxnm 25440 rrxds 25442 rrxmval 25454 rrxmfval 25455 rrxbasefi 25459 rrxmetfi 25461 ehlbase 25464 k0004ss2 44688 rrnprjdstle 46835

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