Theorem dsmmbas2 21775

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 6910	. . . . 5 ⊢ (Base‘𝑃) = (Base‘(𝑆Xs𝑅))
4	3	rabeqi 3447	. . . 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 2999	. . . . . . . 8 ⊢ ((((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) ≠ ((0g ∘ 𝑅)‘𝑥) ↔ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))))
9	8	rabbidva 3440	. . . . . . 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 2735	. . . . . . . . 9 ⊢ (𝑆Xs𝑅) = (𝑆Xs𝑅)
11		eqid 2735	. . . . . . . . 9 ⊢ (Base‘(𝑆Xs𝑅)) = (Base‘(𝑆Xs𝑅))
12		reldmprds 17495	. . . . . . . . . . 11 ⊢ Rel dom Xs
13	10, 11, 12	strov2rcl 17253	. . . . . . . . . 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 17518	. . . . . . . 8 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 Fn 𝐼)
18		fn0g 18689	. . . . . . . . . . . 12 ⊢ 0g Fn V
19		dffn2 6739	. . . . . . . . . . . 12 ⊢ (0g Fn V ↔ 0g:V⟶V)
20	18, 19	mpbi 230	. . . . . . . . . . 11 ⊢ 0g:V⟶V
21		dffn2 6739	. . . . . . . . . . . 12 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
22	21	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑅 Fn 𝐼 → 𝑅:𝐼⟶V)
23		fco 6761	. . . . . . . . . . 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 6738	. . . . . . . . 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 6672	. . . . . . . . 9 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
30	29	rabeqdv 3449	. . . . . . . 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 2785	. . . . . 6 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom (𝑓 ∖ (0g ∘ 𝑅)) = {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))})
33	32	eleq1d 2824	. . . . 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 3440	. . . 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 2787	. . 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 2735	. . . . 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 21773	. . . 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 2775	. 2 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} = (Base‘(𝑆 ⊕m 𝑅)))
41	1, 40	eqtrid 2787	1 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝐵 = (Base‘(𝑆 ⊕m 𝑅)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  {crab 3433  Vcvv 3478   ∖ cdif 3960  dom cdm 5689   ∘ ccom 5693   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Fincfn 8984  Basecbs 17245  0_gc0g 17486  X_scprds 17492   ⊕_m cdsmm 21769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  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 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-prds 17494  df-dsmm 21770

This theorem is referenced by:  dsmmfi  21776  frlmbas  21793