dsmmbase - Metamath Proof Explorer

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Theorem dsmmbase 21948

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 3471	. 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 4030	. . . 4 ⊢ 𝐵 ⊆ (Base‘(𝑆X_s𝑅))
4		eqid 2760	. . . . 5 ⊢ ((𝑆X_s𝑅) ↾_s 𝐵) = ((𝑆X_s𝑅) ↾_s 𝐵)
5		eqid 2760	. . . . 5 ⊢ (Base‘(𝑆X_s𝑅)) = (Base‘(𝑆X_s𝑅))
6	4, 5	ressbas2 17330	. . . 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 21947	. . . 4 ⊢ (𝑅 ∈ V → (𝑆 ⊕_m 𝑅) = ((𝑆X_s𝑅) ↾_s 𝐵))
9	8	fveq2d 6882	. . 3 ⊢ (𝑅 ∈ V → (Base‘(𝑆 ⊕_m 𝑅)) = (Base‘((𝑆X_s𝑅) ↾_s 𝐵)))
10	7, 9	eqtr4id 2814	. 2 ⊢ (𝑅 ∈ V → 𝐵 = (Base‘(𝑆 ⊕_m 𝑅)))
11	1, 10	syl 18	1 ⊢ (𝑅 ∈ 𝑉 → 𝐵 = (Base‘(𝑆 ⊕_m 𝑅)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 {crab 3412 Vcvv 3450 ⊆ wss 3899 dom cdm 5655 ‘cfv 6533 (class class class)co 7413 Fincfn 8952 Basecbs 17301 ↾_s cress 17322 0_gc0g 17524 X_scprds 17530 ⊕_m cdsmm 21944

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-fz 13562 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-hom 17366 df-cco 17367 df-prds 17532 df-dsmm 21945

This theorem is used by: dsmmbas2 21950 dsmmelbas 21952 dsmmsubg 21956

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