Theorem dsmmbas2 21757

Description: Base set of the direct sum module using the fndmin 7065 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 6909	. . . . 5 ⊢ (Base‘𝑃) = (Base‘(𝑆Xs𝑅))
4	3	rabeqi 3450	. . . 4 ⊢ {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin}
5		simpll 767	. . . . . . . . . 10 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑅 Fn 𝐼)
6		fvco2 7006	. . . . . . . . . 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 3001	. . . . . . . 8 ⊢ ((((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) ≠ ((0g ∘ 𝑅)‘𝑥) ↔ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))))
9	8	rabbidva 3443	. . . . . . 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 2737	. . . . . . . . 9 ⊢ (𝑆Xs𝑅) = (𝑆Xs𝑅)
11		eqid 2737	. . . . . . . . 9 ⊢ (Base‘(𝑆Xs𝑅)) = (Base‘(𝑆Xs𝑅))
12		reldmprds 17493	. . . . . . . . . . 11 ⊢ Rel dom Xs
13	10, 11, 12	strov2rcl 17255	. . . . . . . . . 10 ⊢ (𝑓 ∈ (Base‘(𝑆Xs𝑅)) → 𝑆 ∈ V)
14	13	adantl 481	. . . . . . . . 9 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑆 ∈ V)
15		simplr 769	. . . . . . . . 9 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝐼 ∈ 𝑉)
16		simpr 484	. . . . . . . . 9 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 ∈ (Base‘(𝑆Xs𝑅)))
17	10, 11, 14, 15, 5, 16	prdsbasfn 17516	. . . . . . . 8 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 Fn 𝐼)
18		fn0g 18676	. . . . . . . . . . . 12 ⊢ 0g Fn V
19		dffn2 6738	. . . . . . . . . . . 12 ⊢ (0g Fn V ↔ 0g:V⟶V)
20	18, 19	mpbi 230	. . . . . . . . . . 11 ⊢ 0g:V⟶V
21		dffn2 6738	. . . . . . . . . . . 12 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
22	21	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑅 Fn 𝐼 → 𝑅:𝐼⟶V)
23		fco 6760	. . . . . . . . . . 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 6737	. . . . . . . . 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 7062	. . . . . . . 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 6671	. . . . . . . . 9 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
30	29	rabeqdv 3452	. . . . . . . 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 2787	. . . . . 6 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom (𝑓 ∖ (0g ∘ 𝑅)) = {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))})
33	32	eleq1d 2826	. . . . 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 3443	. . . 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 2789	. . 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 7237	. . . 4 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝑅 ∈ V)
37		eqid 2737	. . . . 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 21755	. . . 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 2777	. 2 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} = (Base‘(𝑆 ⊕m 𝑅)))
41	1, 40	eqtrid 2789	1 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝐵 = (Base‘(𝑆 ⊕m 𝑅)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  {crab 3436  Vcvv 3480   ∖ cdif 3948  dom cdm 5685   ∘ ccom 5689   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Fincfn 8985  Basecbs 17247  0_gc0g 17484  X_scprds 17490   ⊕_m cdsmm 21751

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-prds 17492  df-dsmm 21752

This theorem is referenced by:  dsmmfi  21758  frlmbas  21775