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Theorem dsmmbas2 20062
 Description: Base set of the direct sum module using the fndmin 6310 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
dsmmbas2.p 𝑃 = (𝑆Xs𝑅)
dsmmbas2.b 𝐵 = {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin}
Assertion
Ref Expression
dsmmbas2 ((𝑅 Fn 𝐼𝐼𝑉) → 𝐵 = (Base‘(𝑆m 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.
StepHypRef Expression
1 dsmmbas2.b . 2 𝐵 = {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin}
2 dsmmbas2.p . . . . . 6 𝑃 = (𝑆Xs𝑅)
32fveq2i 6181 . . . . 5 (Base‘𝑃) = (Base‘(𝑆Xs𝑅))
4 rabeq 3187 . . . . 5 ((Base‘𝑃) = (Base‘(𝑆Xs𝑅)) → {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin})
53, 4ax-mp 5 . . . 4 {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin}
6 simpll 789 . . . . . . . . . 10 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑅 Fn 𝐼)
7 fvco2 6260 . . . . . . . . . 10 ((𝑅 Fn 𝐼𝑥𝐼) → ((0g𝑅)‘𝑥) = (0g‘(𝑅𝑥)))
86, 7sylan 488 . . . . . . . . 9 ((((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) ∧ 𝑥𝐼) → ((0g𝑅)‘𝑥) = (0g‘(𝑅𝑥)))
98neeq2d 2851 . . . . . . . 8 ((((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) ∧ 𝑥𝐼) → ((𝑓𝑥) ≠ ((0g𝑅)‘𝑥) ↔ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))))
109rabbidva 3183 . . . . . . 7 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → {𝑥𝐼 ∣ (𝑓𝑥) ≠ ((0g𝑅)‘𝑥)} = {𝑥𝐼 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))})
11 eqid 2620 . . . . . . . . 9 (𝑆Xs𝑅) = (𝑆Xs𝑅)
12 eqid 2620 . . . . . . . . 9 (Base‘(𝑆Xs𝑅)) = (Base‘(𝑆Xs𝑅))
13 noel 3911 . . . . . . . . . . . 12 ¬ 𝑓 ∈ ∅
14 reldmprds 16090 . . . . . . . . . . . . . . . 16 Rel dom Xs
1514ovprc1 6669 . . . . . . . . . . . . . . 15 𝑆 ∈ V → (𝑆Xs𝑅) = ∅)
1615fveq2d 6182 . . . . . . . . . . . . . 14 𝑆 ∈ V → (Base‘(𝑆Xs𝑅)) = (Base‘∅))
17 base0 15893 . . . . . . . . . . . . . 14 ∅ = (Base‘∅)
1816, 17syl6eqr 2672 . . . . . . . . . . . . 13 𝑆 ∈ V → (Base‘(𝑆Xs𝑅)) = ∅)
1918eleq2d 2685 . . . . . . . . . . . 12 𝑆 ∈ V → (𝑓 ∈ (Base‘(𝑆Xs𝑅)) ↔ 𝑓 ∈ ∅))
2013, 19mtbiri 317 . . . . . . . . . . 11 𝑆 ∈ V → ¬ 𝑓 ∈ (Base‘(𝑆Xs𝑅)))
2120con4i 113 . . . . . . . . . 10 (𝑓 ∈ (Base‘(𝑆Xs𝑅)) → 𝑆 ∈ V)
2221adantl 482 . . . . . . . . 9 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑆 ∈ V)
23 simplr 791 . . . . . . . . 9 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝐼𝑉)
24 simpr 477 . . . . . . . . 9 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 ∈ (Base‘(𝑆Xs𝑅)))
2511, 12, 22, 23, 6, 24prdsbasfn 16112 . . . . . . . 8 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 Fn 𝐼)
26 fn0g 17243 . . . . . . . . . . . 12 0g Fn V
27 dffn2 6034 . . . . . . . . . . . 12 (0g Fn V ↔ 0g:V⟶V)
2826, 27mpbi 220 . . . . . . . . . . 11 0g:V⟶V
29 dffn2 6034 . . . . . . . . . . . 12 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
3029biimpi 206 . . . . . . . . . . 11 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
31 fco 6045 . . . . . . . . . . 11 ((0g:V⟶V ∧ 𝑅:𝐼⟶V) → (0g𝑅):𝐼⟶V)
3228, 30, 31sylancr 694 . . . . . . . . . 10 (𝑅 Fn 𝐼 → (0g𝑅):𝐼⟶V)
33 ffn 6032 . . . . . . . . . 10 ((0g𝑅):𝐼⟶V → (0g𝑅) Fn 𝐼)
3432, 33syl 17 . . . . . . . . 9 (𝑅 Fn 𝐼 → (0g𝑅) Fn 𝐼)
3534ad2antrr 761 . . . . . . . 8 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → (0g𝑅) Fn 𝐼)
36 fndmdif 6307 . . . . . . . 8 ((𝑓 Fn 𝐼 ∧ (0g𝑅) Fn 𝐼) → dom (𝑓 ∖ (0g𝑅)) = {𝑥𝐼 ∣ (𝑓𝑥) ≠ ((0g𝑅)‘𝑥)})
3725, 35, 36syl2anc 692 . . . . . . 7 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom (𝑓 ∖ (0g𝑅)) = {𝑥𝐼 ∣ (𝑓𝑥) ≠ ((0g𝑅)‘𝑥)})
38 fndm 5978 . . . . . . . . 9 (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
39 rabeq 3187 . . . . . . . . 9 (dom 𝑅 = 𝐼 → {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} = {𝑥𝐼 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))})
4038, 39syl 17 . . . . . . . 8 (𝑅 Fn 𝐼 → {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} = {𝑥𝐼 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))})
4140ad2antrr 761 . . . . . . 7 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} = {𝑥𝐼 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))})
4210, 37, 413eqtr4d 2664 . . . . . 6 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom (𝑓 ∖ (0g𝑅)) = {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))})
4342eleq1d 2684 . . . . 5 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → (dom (𝑓 ∖ (0g𝑅)) ∈ Fin ↔ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin))
4443rabbidva 3183 . . . 4 ((𝑅 Fn 𝐼𝐼𝑉) → {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin})
455, 44syl5eq 2666 . . 3 ((𝑅 Fn 𝐼𝐼𝑉) → {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin})
46 fnex 6466 . . . 4 ((𝑅 Fn 𝐼𝐼𝑉) → 𝑅 ∈ V)
47 eqid 2620 . . . . 5 {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin}
4847dsmmbase 20060 . . . 4 (𝑅 ∈ V → {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin} = (Base‘(𝑆m 𝑅)))
4946, 48syl 17 . . 3 ((𝑅 Fn 𝐼𝐼𝑉) → {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin} = (Base‘(𝑆m 𝑅)))
5045, 49eqtrd 2654 . 2 ((𝑅 Fn 𝐼𝐼𝑉) → {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin} = (Base‘(𝑆m 𝑅)))
511, 50syl5eq 2666 1 ((𝑅 Fn 𝐼𝐼𝑉) → 𝐵 = (Base‘(𝑆m 𝑅)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988   ≠ wne 2791  {crab 2913  Vcvv 3195   ∖ cdif 3564  ∅c0 3907  dom cdm 5104   ∘ ccom 5108   Fn wfn 5871  ⟶wf 5872  ‘cfv 5876  (class class class)co 6635  Fincfn 7940  Basecbs 15838  0gc0g 16081  Xscprds 16087   ⊕m cdsmm 20056 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-mulr 15936  df-sca 15938  df-vsca 15939  df-ip 15940  df-tset 15941  df-ple 15942  df-ds 15945  df-hom 15947  df-cco 15948  df-0g 16083  df-prds 16089  df-dsmm 20057 This theorem is referenced by:  dsmmfi  20063  frlmbas  20080
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