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Theorem frlmelbas 21731

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 2739	. . . . 5 ⊢ {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 } = {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 }
6	2, 3, 4, 5	frlmbas 21730	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 } = (Base‘𝐹))
7	1, 6	eqtr4id 2793	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐵 = {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 })
8	7	eleq2d 2825	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 }))
9		breq1 5075	. . 3 ⊢ (𝑘 = 𝑋 → (𝑘 finSupp 0 ↔ 𝑋 finSupp 0 ))
10	9	elrab 3629	. 2 ⊢ (𝑋 ∈ {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 } ↔ (𝑋 ∈ (𝑁 ↑_m 𝐼) ∧ 𝑋 finSupp 0 ))
11	8, 10	bitrdi 288	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 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 {crab 3391 class class class wbr 5072 ‘cfv 6485 (class class class)co 7356 ↑_m cmap 8763 finSupp cfsupp 9264 Basecbs 17170 0_gc0g 17393 freeLMod cfrlm 21721

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-hom 17235 df-cco 17236 df-0g 17395 df-prds 17401 df-pws 17403 df-sra 21163 df-rgmod 21164 df-dsmm 21707 df-frlm 21722

This theorem is referenced by: frlmbasfsupp 21733 frlmbasmap 21734 frlmsplit2 21748 uvcff 21766 islinds5 33450 islbs5 33463 fedgmullem2 33814 extdgfialglem1 33876 mnringelbased 44661

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