frlmelbas - Metamath Proof Explorer

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Theorem frlmelbas 21651

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 2726	. . . . 5 ⊢ {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 } = {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 }
6	2, 3, 4, 5	frlmbas 21650	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 } = (Base‘𝐹))
7	1, 6	eqtr4id 2785	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐵 = {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 })
8	7	eleq2d 2813	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 }))
9		breq1 5144	. . 3 ⊢ (𝑘 = 𝑋 → (𝑘 finSupp 0 ↔ 𝑋 finSupp 0 ))
10	9	elrab 3678	. 2 ⊢ (𝑋 ∈ {𝑘 ∈ (𝑁 ↑_m 𝐼) ∣ 𝑘 finSupp 0 } ↔ (𝑋 ∈ (𝑁 ↑_m 𝐼) ∧ 𝑋 finSupp 0 ))
11	8, 10	bitrdi 287	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 1533 ∈ wcel 2098 {crab 3426 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 ↑_m cmap 8822 finSupp cfsupp 9363 Basecbs 17153 0_gc0g 17394 freeLMod cfrlm 21641

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-prds 17402 df-pws 17404 df-sra 21021 df-rgmod 21022 df-dsmm 21627 df-frlm 21642

This theorem is referenced by: frlmbasfsupp 21653 frlmbasmap 21654 frlmsplit2 21668 uvcff 21686 islinds5 32986 islbs5 33002 fedgmullem2 33233 mnringelbased 43549

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