Theorem dsmmbas2 21143

Description: Base set of the direct sum module using the fndmin 6995 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 6845	. . . . 5 ⊢ (Base‘𝑃) = (Base‘(𝑆Xs𝑅))
4	3	rabeqi 3420	. . . 4 ⊢ {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin}
5		simpll 765	. . . . . . . . . 10 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑅 Fn 𝐼)
6		fvco2 6938	. . . . . . . . . 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 3414	. . . . . . 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 2736	. . . . . . . . 9 ⊢ (𝑆Xs𝑅) = (𝑆Xs𝑅)
11		eqid 2736	. . . . . . . . 9 ⊢ (Base‘(𝑆Xs𝑅)) = (Base‘(𝑆Xs𝑅))
12		reldmprds 17330	. . . . . . . . . . 11 ⊢ Rel dom Xs
13	10, 11, 12	strov2rcl 17091	. . . . . . . . . 10 ⊢ (𝑓 ∈ (Base‘(𝑆Xs𝑅)) → 𝑆 ∈ V)
14	13	adantl 482	. . . . . . . . 9 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑆 ∈ V)
15		simplr 767	. . . . . . . . 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 17353	. . . . . . . 8 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 Fn 𝐼)
18		fn0g 18518	. . . . . . . . . . . 12 ⊢ 0g Fn V
19		dffn2 6670	. . . . . . . . . . . 12 ⊢ (0g Fn V ↔ 0g:V⟶V)
20	18, 19	mpbi 229	. . . . . . . . . . 11 ⊢ 0g:V⟶V
21		dffn2 6670	. . . . . . . . . . . 12 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
22	21	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑅 Fn 𝐼 → 𝑅:𝐼⟶V)
23		fco 6692	. . . . . . . . . . 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 6669	. . . . . . . . 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 6992	. . . . . . . 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 6605	. . . . . . . . 9 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
30	29	rabeqdv 3422	. . . . . . . 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 2786	. . . . . 6 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom (𝑓 ∖ (0g ∘ 𝑅)) = {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))})
33	32	eleq1d 2822	. . . . 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 3414	. . . 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 2788	. . 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 7167	. . . 4 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝑅 ∈ V)
37		eqid 2736	. . . . 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 21141	. . . 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 2776	. 2 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} = (Base‘(𝑆 ⊕m 𝑅)))
41	1, 40	eqtrid 2788	1 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝐵 = (Base‘(𝑆 ⊕m 𝑅)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  {crab 3407  Vcvv 3445   ∖ cdif 3907  dom cdm 5633   ∘ ccom 5637   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Fincfn 8883  Basecbs 17083  0_gc0g 17321  X_scprds 17327   ⊕_m cdsmm 21137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-prds 17329  df-dsmm 21138

This theorem is referenced by:  dsmmfi  21144  frlmbas  21161