Theorem dsmmbas2 22073

Description: Base set of the direct sum module using the fndmin 7067 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 6911	. . . . 5 ⊢ (Base‘𝑃) = (Base‘(𝑆Xs𝑅))
4	3	rabeqi 3429	. . . 4 ⊢ {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin}
5		simpll 779	. . . . . . . . . 10 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑅 Fn 𝐼)
6		fvco2 7005	. . . . . . . . . 10 ⊢ ((𝑅 Fn 𝐼 ∧ 𝑥 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑥) = (0g‘(𝑅‘𝑥)))
7	5, 6	sylan 592	. . . . . . . . 9 ⊢ ((((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) ∧ 𝑥 ∈ 𝐼) → ((0g ∘ 𝑅)‘𝑥) = (0g‘(𝑅‘𝑥)))
8	7	neeq2d 3019	. . . . . . . 8 ⊢ ((((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥) ≠ ((0g ∘ 𝑅)‘𝑥) ↔ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))))
9	8	rabbidva 3422	. . . . . . 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 2763	. . . . . . . . 9 ⊢ (𝑆Xs𝑅) = (𝑆Xs𝑅)
11		eqid 2763	. . . . . . . . 9 ⊢ (Base‘(𝑆Xs𝑅)) = (Base‘(𝑆Xs𝑅))
12		reldmprds 17648	. . . . . . . . . . 11 ⊢ Rel dom Xs
13	10, 11, 12	strov2rcl 17424	. . . . . . . . . 10 ⊢ (𝑓 ∈ (Base‘(𝑆Xs𝑅)) → 𝑆 ∈ V)
14	13	adantl 487	. . . . . . . . 9 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑆 ∈ V)
15		simplr 781	. . . . . . . . 9 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝐼 ∈ 𝑉)
16		simpr 490	. . . . . . . . 9 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 ∈ (Base‘(𝑆Xs𝑅)))
17	10, 11, 14, 15, 5, 16	prdsbasfn 17671	. . . . . . . 8 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 Fn 𝐼)
18		fn0g 18872	. . . . . . . . . . . 12 ⊢ 0g Fn V
19		dffn2 6733	. . . . . . . . . . . 12 ⊢ (0g Fn V ↔ 0g:V⟶V)
20	18, 19	mpbi 233	. . . . . . . . . . 11 ⊢ 0g:V⟶V
21		dffn2 6733	. . . . . . . . . . . 12 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
22	21	biimpi 219	. . . . . . . . . . 11 ⊢ (𝑅 Fn 𝐼 → 𝑅:𝐼⟶V)
23		fco 6756	. . . . . . . . . . 11 ⊢ ((0g:V⟶V ∧ 𝑅:𝐼⟶V) → (0g ∘ 𝑅):𝐼⟶V)
24	20, 22, 23	sylancr 599	. . . . . . . . . 10 ⊢ (𝑅 Fn 𝐼 → (0g ∘ 𝑅):𝐼⟶V)
25	24	ffnd 6732	. . . . . . . . 9 ⊢ (𝑅 Fn 𝐼 → (0g ∘ 𝑅) Fn 𝐼)
26	5, 25	syl 18	. . . . . . . 8 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → (0g ∘ 𝑅) Fn 𝐼)
27		fndmdif 7064	. . . . . . . 8 ⊢ ((𝑓 Fn 𝐼 ∧ (0g ∘ 𝑅) Fn 𝐼) → dom (𝑓 ∖ (0g ∘ 𝑅)) = {𝑥 ∈ 𝐼 ∣ (𝑓‘𝑥) ≠ ((0g ∘ 𝑅)‘𝑥)})
28	17, 26, 27	syl2anc 596	. . . . . . 7 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom (𝑓 ∖ (0g ∘ 𝑅)) = {𝑥 ∈ 𝐼 ∣ (𝑓‘𝑥) ≠ ((0g ∘ 𝑅)‘𝑥)})
29		fndm 6664	. . . . . . . . 9 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
30	29	rabeqdv 3431	. . . . . . . 8 ⊢ (𝑅 Fn 𝐼 → {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} = {𝑥 ∈ 𝐼 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))})
31	5, 30	syl 18	. . . . . . 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 2808	. . . . . 6 ⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom (𝑓 ∖ (0g ∘ 𝑅)) = {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))})
33	32	eleq1d 2849	. . . . 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 3422	. . . 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 2810	. . 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 7246	. . . 4 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝑅 ∈ V)
37		eqid 2763	. . . . 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 22071	. . . 4 ⊢ (𝑅 ∈ V → {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} = (Base‘(𝑆 ⊕m 𝑅)))
39	36, 38	syl 18	. . 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 2798	. 2 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} = (Base‘(𝑆 ⊕m 𝑅)))
41	1, 40	eqtrid 2810	1 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝐵 = (Base‘(𝑆 ⊕m 𝑅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  {crab 3416  Vcvv 3454   ∖ cdif 4048  dom cdm 5655   ∘ ccom 5659   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443  Fincfn 9001  Basecbs 17416  0_gc0g 17639  X_scprds 17645   ⊕_m cdsmm 22067

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-map 8877  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-sup 9461  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-mulr 17471  df-sca 17473  df-vsca 17474  df-ip 17475  df-tset 17476  df-ple 17477  df-ds 17479  df-hom 17481  df-cco 17482  df-0g 17641  df-prds 17647  df-dsmm 22068

This theorem is used by:  dsmmfi  22074  frlmbas  22091