dsmmbase - Metamath Proof Explorer

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Theorem dsmmbase 21935

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 3478	. 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 4037	. . . 4 ⊢ 𝐵 ⊆ (Base‘(𝑆X_s𝑅))
4		eqid 2765	. . . . 5 ⊢ ((𝑆X_s𝑅) ↾_s 𝐵) = ((𝑆X_s𝑅) ↾_s 𝐵)
5		eqid 2765	. . . . 5 ⊢ (Base‘(𝑆X_s𝑅)) = (Base‘(𝑆X_s𝑅))
6	4, 5	ressbas2 17320	. . . 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 21934	. . . 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 2819	. 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 2146 ≠ wne 2960 {crab 3418 Vcvv 3457 ⊆ wss 3906 dom cdm 5663 ‘cfv 6540 (class class class)co 7419 Fincfn 8949 Basecbs 17291 ↾_s cress 17312 0_gc0g 17514 X_scprds 17520 ⊕_m cdsmm 21931

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-fz 13552 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-hom 17356 df-cco 17357 df-prds 17522 df-dsmm 21932

This theorem is used by: dsmmbas2 21937 dsmmelbas 21939 dsmmsubg 21943

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