dsmmbase - Metamath Proof Explorer

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Theorem dsmmbase 21630

Description: Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)

**Hypothesis**
Ref	Expression
dsmmval.b	⊢ 𝐵 = {𝑓 ∈ (Base‘(𝑆X_s𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0_g‘(𝑅‘𝑥))} ∈ Fin}

**Assertion**
Ref	Expression
dsmmbase	⊢ (𝑅 ∈ 𝑉 → 𝐵 = (Base‘(𝑆 ⊕_m 𝑅)))

Distinct variable groups: 𝑆,𝑓,𝑥 𝑅,𝑓,𝑥 Allowed substitution hints: 𝐵(𝑥,𝑓) 𝑉(𝑥,𝑓)

**Proof of Theorem dsmmbase**
Step	Hyp	Ref	Expression
1		elex 3487	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		dsmmval.b	. . . . 5 ⊢ 𝐵 = {𝑓 ∈ (Base‘(𝑆X_s𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0_g‘(𝑅‘𝑥))} ∈ Fin}
3	2	ssrab3 4075	. . . 4 ⊢ 𝐵 ⊆ (Base‘(𝑆X_s𝑅))
4		eqid 2726	. . . . 5 ⊢ ((𝑆X_s𝑅) ↾_s 𝐵) = ((𝑆X_s𝑅) ↾_s 𝐵)
5		eqid 2726	. . . . 5 ⊢ (Base‘(𝑆X_s𝑅)) = (Base‘(𝑆X_s𝑅))
6	4, 5	ressbas2 17191	. . . 4 ⊢ (𝐵 ⊆ (Base‘(𝑆X_s𝑅)) → 𝐵 = (Base‘((𝑆X_s𝑅) ↾_s 𝐵)))
7	3, 6	ax-mp 5	. . 3 ⊢ 𝐵 = (Base‘((𝑆X_s𝑅) ↾_s 𝐵))
8	2	dsmmval 21629	. . . 4 ⊢ (𝑅 ∈ V → (𝑆 ⊕_m 𝑅) = ((𝑆X_s𝑅) ↾_s 𝐵))
9	8	fveq2d 6889	. . 3 ⊢ (𝑅 ∈ V → (Base‘(𝑆 ⊕_m 𝑅)) = (Base‘((𝑆X_s𝑅) ↾_s 𝐵)))
10	7, 9	eqtr4id 2785	. 2 ⊢ (𝑅 ∈ V → 𝐵 = (Base‘(𝑆 ⊕_m 𝑅)))
11	1, 10	syl 17	1 ⊢ (𝑅 ∈ 𝑉 → 𝐵 = (Base‘(𝑆 ⊕_m 𝑅)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 {crab 3426 Vcvv 3468 ⊆ wss 3943 dom cdm 5669 ‘cfv 6537 (class class class)co 7405 Fincfn 8941 Basecbs 17153 ↾_s cress 17182 0_gc0g 17394 X_scprds 17400 ⊕_m cdsmm 21626

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-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-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-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-prds 17402 df-dsmm 21627

This theorem is referenced by: dsmmbas2 21632 dsmmelbas 21634 dsmmsubg 21638

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