frlmelbas - Metamath Proof Explorer

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Theorem frlmelbas 20873

Description: Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.)

**Hypotheses**
Ref	Expression
frlmval.f	⊢ 𝐹 = (𝑅 freeLMod 𝐼)
frlmelbas.n	⊢ 𝑁 = (Base‘𝑅)
frlmelbas.z	⊢ 0 = (0_g‘𝑅)
frlmelbas.b	⊢ 𝐵 = (Base‘𝐹)

**Assertion**
Ref	Expression
frlmelbas	⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (𝑁 ↑_m 𝐼) ∧ 𝑋 finSupp 0 )))

Proof of Theorem frlmelbas Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frlmelbas.b	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
2		frlmval.f	. . . . 5 ⊢ 𝐹 = (𝑅 freeLMod 𝐼)
3		frlmelbas.n	. . . . 5 ⊢ 𝑁 = (Base‘𝑅)
4		frlmelbas.z	. . . . 5 ⊢ 0 = (0_g‘𝑅)
5		eqid 2738	. . . . 5 ⊢ {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 } = {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 }
6	2, 3, 4, 5	frlmbas 20872	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 } = (Base‘𝐹))
7	1, 6	eqtr4id 2798	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐵 = {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 })
8	7	eleq2d 2824	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 }))
9		breq1 5073	. . 3 ⊢ (𝑘 = 𝑋 → (𝑘 finSupp 0 ↔ 𝑋 finSupp 0 ))
10	9	elrab 3617	. 2 ⊢ (𝑋 ∈ {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 } ↔ (𝑋 ∈ (𝑁 ↑_m 𝐼) ∧ 𝑋 finSupp 0 ))
11	8, 10	bitrdi 286	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (𝑁 ↑_m 𝐼) ∧ 𝑋 finSupp 0 )))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 {crab 3067 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 finSupp cfsupp 9058 Basecbs 16840 0_gc0g 17067 freeLMod cfrlm 20863

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-hom 16912 df-cco 16913 df-0g 17069 df-prds 17075 df-pws 17077 df-sra 20349 df-rgmod 20350 df-dsmm 20849 df-frlm 20864

This theorem is referenced by: frlmbasfsupp 20875 frlmbasmap 20876 frlmsplit2 20890 uvcff 20908 islinds5 31465 fedgmullem2 31613 mnringelbased 41721

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