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Theorem dsmmbas2 19900
Description: Base set of the direct sum module using the fndmin 6232 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 6106 . . . . 5 (Base‘𝑃) = (Base‘(𝑆Xs𝑅))
4 rabeq 3166 . . . . 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 786 . . . . . . . . . 10 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑅 Fn 𝐼)
7 fvco2 6183 . . . . . . . . . 10 ((𝑅 Fn 𝐼𝑥𝐼) → ((0g𝑅)‘𝑥) = (0g‘(𝑅𝑥)))
86, 7sylan 487 . . . . . . . . 9 ((((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) ∧ 𝑥𝐼) → ((0g𝑅)‘𝑥) = (0g‘(𝑅𝑥)))
98neeq2d 2842 . . . . . . . 8 ((((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) ∧ 𝑥𝐼) → ((𝑓𝑥) ≠ ((0g𝑅)‘𝑥) ↔ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))))
109rabbidva 3163 . . . . . . 7 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → {𝑥𝐼 ∣ (𝑓𝑥) ≠ ((0g𝑅)‘𝑥)} = {𝑥𝐼 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))})
11 eqid 2610 . . . . . . . . 9 (𝑆Xs𝑅) = (𝑆Xs𝑅)
12 eqid 2610 . . . . . . . . 9 (Base‘(𝑆Xs𝑅)) = (Base‘(𝑆Xs𝑅))
13 noel 3878 . . . . . . . . . . . 12 ¬ 𝑓 ∈ ∅
14 reldmprds 15932 . . . . . . . . . . . . . . . 16 Rel dom Xs
1514ovprc1 6582 . . . . . . . . . . . . . . 15 𝑆 ∈ V → (𝑆Xs𝑅) = ∅)
1615fveq2d 6107 . . . . . . . . . . . . . 14 𝑆 ∈ V → (Base‘(𝑆Xs𝑅)) = (Base‘∅))
17 base0 15740 . . . . . . . . . . . . . 14 ∅ = (Base‘∅)
1816, 17syl6eqr 2662 . . . . . . . . . . . . 13 𝑆 ∈ V → (Base‘(𝑆Xs𝑅)) = ∅)
1918eleq2d 2673 . . . . . . . . . . . 12 𝑆 ∈ V → (𝑓 ∈ (Base‘(𝑆Xs𝑅)) ↔ 𝑓 ∈ ∅))
2013, 19mtbiri 316 . . . . . . . . . . 11 𝑆 ∈ V → ¬ 𝑓 ∈ (Base‘(𝑆Xs𝑅)))
2120con4i 112 . . . . . . . . . 10 (𝑓 ∈ (Base‘(𝑆Xs𝑅)) → 𝑆 ∈ V)
2221adantl 481 . . . . . . . . 9 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑆 ∈ V)
23 simplr 788 . . . . . . . . 9 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝐼𝑉)
24 simpr 476 . . . . . . . . 9 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 ∈ (Base‘(𝑆Xs𝑅)))
2511, 12, 22, 23, 6, 24prdsbasfn 15954 . . . . . . . 8 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 Fn 𝐼)
26 fn0g 17085 . . . . . . . . . . . 12 0g Fn V
27 dffn2 5960 . . . . . . . . . . . 12 (0g Fn V ↔ 0g:V⟶V)
2826, 27mpbi 219 . . . . . . . . . . 11 0g:V⟶V
29 dffn2 5960 . . . . . . . . . . . 12 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
3029biimpi 205 . . . . . . . . . . 11 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
31 fco 5971 . . . . . . . . . . 11 ((0g:V⟶V ∧ 𝑅:𝐼⟶V) → (0g𝑅):𝐼⟶V)
3228, 30, 31sylancr 694 . . . . . . . . . 10 (𝑅 Fn 𝐼 → (0g𝑅):𝐼⟶V)
33 ffn 5958 . . . . . . . . . 10 ((0g𝑅):𝐼⟶V → (0g𝑅) Fn 𝐼)
3432, 33syl 17 . . . . . . . . 9 (𝑅 Fn 𝐼 → (0g𝑅) Fn 𝐼)
3534ad2antrr 758 . . . . . . . 8 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → (0g𝑅) Fn 𝐼)
36 fndmdif 6229 . . . . . . . 8 ((𝑓 Fn 𝐼 ∧ (0g𝑅) Fn 𝐼) → dom (𝑓 ∖ (0g𝑅)) = {𝑥𝐼 ∣ (𝑓𝑥) ≠ ((0g𝑅)‘𝑥)})
3725, 35, 36syl2anc 691 . . . . . . 7 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom (𝑓 ∖ (0g𝑅)) = {𝑥𝐼 ∣ (𝑓𝑥) ≠ ((0g𝑅)‘𝑥)})
38 fndm 5904 . . . . . . . . 9 (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
39 rabeq 3166 . . . . . . . . 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 758 . . . . . . 7 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} = {𝑥𝐼 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))})
4210, 37, 413eqtr4d 2654 . . . . . 6 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom (𝑓 ∖ (0g𝑅)) = {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))})
4342eleq1d 2672 . . . . 5 (((𝑅 Fn 𝐼𝐼𝑉) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → (dom (𝑓 ∖ (0g𝑅)) ∈ Fin ↔ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin))
4443rabbidva 3163 . . . 4 ((𝑅 Fn 𝐼𝐼𝑉) → {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin})
455, 44syl5eq 2656 . . 3 ((𝑅 Fn 𝐼𝐼𝑉) → {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin})
46 fnex 6386 . . . 4 ((𝑅 Fn 𝐼𝐼𝑉) → 𝑅 ∈ V)
47 eqid 2610 . . . . 5 {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin}
4847dsmmbase 19898 . . . 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 2644 . 2 ((𝑅 Fn 𝐼𝐼𝑉) → {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin} = (Base‘(𝑆m 𝑅)))
511, 50syl5eq 2656 1 ((𝑅 Fn 𝐼𝐼𝑉) → 𝐵 = (Base‘(𝑆m 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  wne 2780  {crab 2900  Vcvv 3173  cdif 3537  c0 3874  dom cdm 5038  ccom 5042   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  Fincfn 7841  Basecbs 15695  0gc0g 15923  Xscprds 15929  m cdsmm 19894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-prds 15931  df-dsmm 19895
This theorem is referenced by:  dsmmfi  19901  frlmbas  19918
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