Theorem dsmmbas2 20944

Description: Base set of the direct sum module using the fndmin 6922 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

Ref	Expression
dsmmbas2.p	⊢ 𝑃 = (𝑆Xs𝑅)
dsmmbas2.b	⊢ 𝐵 = {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin}

Assertion

Ref	Expression
dsmmbas2	⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝐵 = (Base‘(𝑆 ⊕m 𝑅)))

Distinct variable groups:   𝑆,𝑓   𝑅,𝑓   𝑃,𝑓   𝑓,𝐼   𝑓,𝑉

Allowed substitution hint:   𝐵(𝑓)

Proof of Theorem dsmmbas2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dsmmbas2.b	. 2 ⊢ 𝐵 = {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin}
2		dsmmbas2.p	. . . . . 6 ⊢ 𝑃 = (𝑆Xs𝑅)
3	2	fveq2i 6777	. . . . 5 ⊢ (Base‘𝑃) = (Base‘(𝑆Xs𝑅))
4	3	rabeqi 3416	. . . 4 ⊢ {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin}
5		simpll 764	. . . . . . . . . 10 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑅 Fn 𝐼)
6		fvco2 6865	. . . . . . . . . 10 ⊢ ((𝑅 Fn 𝐼 ∧ 𝑥 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑥) = (0g‘(𝑅‘𝑥)))
7	5, 6	sylan 580	. . . . . . . . 9 ⊢ ((((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) ∧ 𝑥 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑥) = (0g‘(𝑅‘𝑥)))
8	7	neeq2d 3004	. . . . . . . 8 ⊢ ((((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) ≠ ((0g ∘ 𝑅)‘𝑥) ↔ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))))
9	8	rabbidva 3413	. . . . . . 7 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → {𝑥 ∈ 𝐼 ∣ (𝑓‘𝑥) ≠ ((0g ∘ 𝑅)‘𝑥)} = {𝑥 ∈ 𝐼 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))})
10		eqid 2738	. . . . . . . . 9 ⊢ (𝑆Xs𝑅) = (𝑆Xs𝑅)
11		eqid 2738	. . . . . . . . 9 ⊢ (Base‘(𝑆Xs𝑅)) = (Base‘(𝑆Xs𝑅))
12		reldmprds 17159	. . . . . . . . . . 11 ⊢ Rel dom Xs
13	10, 11, 12	strov2rcl 16920	. . . . . . . . . 10 ⊢ (𝑓 ∈ (Base‘(𝑆Xs𝑅)) → 𝑆 ∈ V)
14	13	adantl 482	. . . . . . . . 9 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑆 ∈ V)
15		simplr 766	. . . . . . . . 9 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝐼 ∈ 𝑉)
16		simpr 485	. . . . . . . . 9 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 ∈ (Base‘(𝑆Xs𝑅)))
17	10, 11, 14, 15, 5, 16	prdsbasfn 17182	. . . . . . . 8 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 Fn 𝐼)
18		fn0g 18347	. . . . . . . . . . . 12 ⊢ 0g Fn V
19		dffn2 6602	. . . . . . . . . . . 12 ⊢ (0g Fn V ↔ 0g:V⟶V)
20	18, 19	mpbi 229	. . . . . . . . . . 11 ⊢ 0g:V⟶V
21		dffn2 6602	. . . . . . . . . . . 12 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
22	21	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑅 Fn 𝐼 → 𝑅:𝐼⟶V)
23		fco 6624	. . . . . . . . . . 11 ⊢ ((0g:V⟶V ∧ 𝑅:𝐼⟶V) → (0g ∘ 𝑅):𝐼⟶V)
24	20, 22, 23	sylancr 587	. . . . . . . . . 10 ⊢ (𝑅 Fn 𝐼 → (0g ∘ 𝑅):𝐼⟶V)
25	24	ffnd 6601	. . . . . . . . 9 ⊢ (𝑅 Fn 𝐼 → (0g ∘ 𝑅) Fn 𝐼)
26	5, 25	syl 17	. . . . . . . 8 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → (0g ∘ 𝑅) Fn 𝐼)
27		fndmdif 6919	. . . . . . . 8 ⊢ ((𝑓 Fn 𝐼 ∧ (0g ∘ 𝑅) Fn 𝐼) → dom (𝑓 ∖ (0g ∘ 𝑅)) = {𝑥 ∈ 𝐼 ∣ (𝑓‘𝑥) ≠ ((0g ∘ 𝑅)‘𝑥)})
28	17, 26, 27	syl2anc 584	. . . . . . 7 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom (𝑓 ∖ (0g ∘ 𝑅)) = {𝑥 ∈ 𝐼 ∣ (𝑓‘𝑥) ≠ ((0g ∘ 𝑅)‘𝑥)})
29		fndm 6536	. . . . . . . . 9 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
30	29	rabeqdv 3419	. . . . . . . 8 ⊢ (𝑅 Fn 𝐼 → {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} = {𝑥 ∈ 𝐼 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))})
31	5, 30	syl 17	. . . . . . 7 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} = {𝑥 ∈ 𝐼 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))})
32	9, 28, 31	3eqtr4d 2788	. . . . . 6 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom (𝑓 ∖ (0g ∘ 𝑅)) = {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))})
33	32	eleq1d 2823	. . . . 5 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → (dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin ↔ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin))
34	33	rabbidva 3413	. . . 4 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin})
35	4, 34	eqtrid 2790	. . 3 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin})
36		fnex 7093	. . . 4 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝑅 ∈ V)
37		eqid 2738	. . . . 5 ⊢ {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin}
38	37	dsmmbase 20942	. . . 4 ⊢ (𝑅 ∈ V → {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} = (Base‘(𝑆 ⊕m 𝑅)))
39	36, 38	syl 17	. . 3 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} = (Base‘(𝑆 ⊕m 𝑅)))
40	35, 39	eqtrd 2778	. 2 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} = (Base‘(𝑆 ⊕m 𝑅)))
41	1, 40	eqtrid 2790	1 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝐵 = (Base‘(𝑆 ⊕m 𝑅)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {crab 3068  Vcvv 3432   ∖ cdif 3884  dom cdm 5589   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  Basecbs 16912  0_gc0g 17150  X_scprds 17156   ⊕_m cdsmm 20938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-hom 16986  df-cco 16987  df-0g 17152  df-prds 17158  df-dsmm 20939

This theorem is referenced by:  dsmmfi  20945  frlmbas  20962