dsmmbase - Metamath Proof Explorer

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Theorem dsmmbase 21642

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 3457	. 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 4033	. . . 4 ⊢ 𝐵 ⊆ (Base‘(𝑆X_s𝑅))
4		eqid 2729	. . . . 5 ⊢ ((𝑆X_s𝑅) ↾_s 𝐵) = ((𝑆X_s𝑅) ↾_s 𝐵)
5		eqid 2729	. . . . 5 ⊢ (Base‘(𝑆X_s𝑅)) = (Base‘(𝑆X_s𝑅))
6	4, 5	ressbas2 17149	. . . 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 21641	. . . 4 ⊢ (𝑅 ∈ V → (𝑆 ⊕_m 𝑅) = ((𝑆X_s𝑅) ↾_s 𝐵))
9	8	fveq2d 6826	. . 3 ⊢ (𝑅 ∈ V → (Base‘(𝑆 ⊕_m 𝑅)) = (Base‘((𝑆X_s𝑅) ↾_s 𝐵)))
10	7, 9	eqtr4id 2783	. 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 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3394 Vcvv 3436 ⊆ wss 3903 dom cdm 5619 ‘cfv 6482 (class class class)co 7349 Fincfn 8872 Basecbs 17120 ↾_s cress 17141 0_gc0g 17343 X_scprds 17349 ⊕_m cdsmm 21638

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-fz 13411 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-hom 17185 df-cco 17186 df-prds 17351 df-dsmm 21639

This theorem is referenced by: dsmmbas2 21644 dsmmelbas 21646 dsmmsubg 21650

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