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Theorem pwslnmlem2 36479
Description: A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypotheses
Ref Expression
pwslnmlem2.a 𝐴 ∈ V
pwslnmlem2.b 𝐵 ∈ V
pwslnmlem2.x 𝑋 = (𝑊s 𝐴)
pwslnmlem2.y 𝑌 = (𝑊s 𝐵)
pwslnmlem2.z 𝑍 = (𝑊s (𝐴𝐵))
pwslnmlem2.w (𝜑𝑊 ∈ LMod)
pwslnmlem2.dj (𝜑 → (𝐴𝐵) = ∅)
pwslnmlem2.xn (𝜑𝑋 ∈ LNoeM)
pwslnmlem2.yn (𝜑𝑌 ∈ LNoeM)
Assertion
Ref Expression
pwslnmlem2 (𝜑𝑍 ∈ LNoeM)

Proof of Theorem pwslnmlem2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwslnmlem2.w . . 3 (𝜑𝑊 ∈ LMod)
2 pwslnmlem2.a . . . . 5 𝐴 ∈ V
3 pwslnmlem2.b . . . . 5 𝐵 ∈ V
42, 3unex 6826 . . . 4 (𝐴𝐵) ∈ V
54a1i 11 . . 3 (𝜑 → (𝐴𝐵) ∈ V)
6 ssun1 3732 . . . 4 𝐴 ⊆ (𝐴𝐵)
76a1i 11 . . 3 (𝜑𝐴 ⊆ (𝐴𝐵))
8 pwslnmlem2.z . . . 4 𝑍 = (𝑊s (𝐴𝐵))
9 pwslnmlem2.x . . . 4 𝑋 = (𝑊s 𝐴)
10 eqid 2604 . . . 4 (Base‘𝑍) = (Base‘𝑍)
11 eqid 2604 . . . 4 (Base‘𝑋) = (Base‘𝑋)
12 eqid 2604 . . . 4 (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) = (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))
138, 9, 10, 11, 12pwssplit3 18823 . . 3 ((𝑊 ∈ LMod ∧ (𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) ∈ (𝑍 LMHom 𝑋))
141, 5, 7, 13syl3anc 1317 . 2 (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) ∈ (𝑍 LMHom 𝑋))
15 fvex 6093 . . . . . 6 (0g𝑋) ∈ V
1612mptiniseg 5527 . . . . . 6 ((0g𝑋) ∈ V → ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (0g𝑋)})
1715, 16ax-mp 5 . . . . 5 ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (0g𝑋)}
18 lmodgrp 18634 . . . . . . . . . 10 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
19 grpmnd 17193 . . . . . . . . . 10 (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
201, 18, 193syl 18 . . . . . . . . 9 (𝜑𝑊 ∈ Mnd)
21 eqid 2604 . . . . . . . . . 10 (0g𝑊) = (0g𝑊)
229, 21pws0g 17090 . . . . . . . . 9 ((𝑊 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × {(0g𝑊)}) = (0g𝑋))
2320, 2, 22sylancl 692 . . . . . . . 8 (𝜑 → (𝐴 × {(0g𝑊)}) = (0g𝑋))
2423eqcomd 2610 . . . . . . 7 (𝜑 → (0g𝑋) = (𝐴 × {(0g𝑊)}))
2524eqeq2d 2614 . . . . . 6 (𝜑 → ((𝑥𝐴) = (0g𝑋) ↔ (𝑥𝐴) = (𝐴 × {(0g𝑊)})))
2625rabbidv 3158 . . . . 5 (𝜑 → {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (0g𝑋)} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
2717, 26syl5eq 2650 . . . 4 (𝜑 → ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
2827oveq2d 6538 . . 3 (𝜑 → (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) = (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}))
29 pwslnmlem2.yn . . . 4 (𝜑𝑌 ∈ LNoeM)
30 pwslnmlem2.dj . . . . . 6 (𝜑 → (𝐴𝐵) = ∅)
31 eqid 2604 . . . . . . 7 {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}
32 eqid 2604 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵)) = (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵))
33 pwslnmlem2.y . . . . . . 7 𝑌 = (𝑊s 𝐵)
34 eqid 2604 . . . . . . 7 (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) = (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
358, 10, 21, 31, 32, 9, 33, 34pwssplit4 36475 . . . . . 6 ((𝑊 ∈ LMod ∧ (𝐴𝐵) ∈ V ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵)) ∈ ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) LMIso 𝑌))
361, 5, 30, 35syl3anc 1317 . . . . 5 (𝜑 → (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵)) ∈ ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) LMIso 𝑌))
37 brlmici 18831 . . . . 5 ((𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵)) ∈ ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) LMIso 𝑌) → (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ≃𝑚 𝑌)
38 lnmlmic 36474 . . . . 5 ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ≃𝑚 𝑌 → ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ∈ LNoeM ↔ 𝑌 ∈ LNoeM))
3936, 37, 383syl 18 . . . 4 (𝜑 → ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ∈ LNoeM ↔ 𝑌 ∈ LNoeM))
4029, 39mpbird 245 . . 3 (𝜑 → (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ∈ LNoeM)
4128, 40eqeltrd 2682 . 2 (𝜑 → (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) ∈ LNoeM)
428, 9, 10, 11, 12pwssplit1 18821 . . . . . . 7 ((𝑊 ∈ Mnd ∧ (𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
4320, 5, 7, 42syl3anc 1317 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
44 forn 6011 . . . . . 6 ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)):(Base‘𝑍)–onto→(Base‘𝑋) → ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) = (Base‘𝑋))
4543, 44syl 17 . . . . 5 (𝜑 → ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) = (Base‘𝑋))
4645oveq2d 6538 . . . 4 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = (𝑋s (Base‘𝑋)))
47 pwslnmlem2.xn . . . . 5 (𝜑𝑋 ∈ LNoeM)
4811ressid 15703 . . . . 5 (𝑋 ∈ LNoeM → (𝑋s (Base‘𝑋)) = 𝑋)
4947, 48syl 17 . . . 4 (𝜑 → (𝑋s (Base‘𝑋)) = 𝑋)
5046, 49eqtrd 2638 . . 3 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = 𝑋)
5150, 47eqeltrd 2682 . 2 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) ∈ LNoeM)
52 eqid 2604 . . 3 (0g𝑋) = (0g𝑋)
53 eqid 2604 . . 3 ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})
54 eqid 2604 . . 3 (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) = (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}))
55 eqid 2604 . . 3 (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)))
5652, 53, 54, 55lmhmlnmsplit 36473 . 2 (((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) ∈ (𝑍 LMHom 𝑋) ∧ (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) ∈ LNoeM ∧ (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) ∈ LNoeM) → 𝑍 ∈ LNoeM)
5714, 41, 51, 56syl3anc 1317 1 (𝜑𝑍 ∈ LNoeM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1975  {crab 2894  Vcvv 3167  cun 3532  cin 3533  wss 3534  c0 3868  {csn 4119   class class class wbr 4572  cmpt 4632   × cxp 5021  ccnv 5022  ran crn 5024  cres 5025  cima 5026  ontowfo 5783  cfv 5785  (class class class)co 6522  Basecbs 15636  s cress 15637  0gc0g 15864  s cpws 15871  Mndcmnd 17058  Grpcgrp 17186  LModclmod 18627   LMHom clmhm 18781   LMIso clmim 18782  𝑚 clmic 18783  LNoeMclnm 36461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-cnex 9843  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-pre-mulgt0 9864
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-of 6767  df-om 6930  df-1st 7031  df-2nd 7032  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-oadd 7423  df-er 7601  df-map 7718  df-ixp 7767  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-sup 8203  df-pnf 9927  df-mnf 9928  df-xr 9929  df-ltxr 9930  df-le 9931  df-sub 10114  df-neg 10115  df-nn 10863  df-2 10921  df-3 10922  df-4 10923  df-5 10924  df-6 10925  df-7 10926  df-8 10927  df-9 10928  df-n0 11135  df-z 11206  df-dec 11321  df-uz 11515  df-fz 12148  df-struct 15638  df-ndx 15639  df-slot 15640  df-base 15641  df-sets 15642  df-ress 15643  df-plusg 15722  df-mulr 15723  df-sca 15725  df-vsca 15726  df-ip 15727  df-tset 15728  df-ple 15729  df-ds 15732  df-hom 15734  df-cco 15735  df-0g 15866  df-prds 15872  df-pws 15874  df-mgm 17006  df-sgrp 17048  df-mnd 17059  df-submnd 17100  df-grp 17189  df-minusg 17190  df-sbg 17191  df-subg 17355  df-ghm 17422  df-cntz 17514  df-lsm 17815  df-cmn 17959  df-abl 17960  df-mgp 18254  df-ur 18266  df-ring 18313  df-lmod 18629  df-lss 18695  df-lsp 18734  df-lmhm 18784  df-lmim 18785  df-lmic 18786  df-lfig 36454  df-lnm 36462
This theorem is referenced by:  pwslnm  36480
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