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Theorem pwslnmlem2 37482
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 6941 . . . 4 (𝐴𝐵) ∈ V
54a1i 11 . . 3 (𝜑 → (𝐴𝐵) ∈ V)
6 ssun1 3768 . . . 4 𝐴 ⊆ (𝐴𝐵)
76a1i 11 . . 3 (𝜑𝐴 ⊆ (𝐴𝐵))
8 pwslnmlem2.z . . . 4 𝑍 = (𝑊s (𝐴𝐵))
9 pwslnmlem2.x . . . 4 𝑋 = (𝑊s 𝐴)
10 eqid 2620 . . . 4 (Base‘𝑍) = (Base‘𝑍)
11 eqid 2620 . . . 4 (Base‘𝑋) = (Base‘𝑋)
12 eqid 2620 . . . 4 (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) = (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))
138, 9, 10, 11, 12pwssplit3 19042 . . 3 ((𝑊 ∈ LMod ∧ (𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) ∈ (𝑍 LMHom 𝑋))
141, 5, 7, 13syl3anc 1324 . 2 (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) ∈ (𝑍 LMHom 𝑋))
15 fvex 6188 . . . . . 6 (0g𝑋) ∈ V
1612mptiniseg 5617 . . . . . 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 18851 . . . . . . . . . 10 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
19 grpmnd 17410 . . . . . . . . . 10 (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
201, 18, 193syl 18 . . . . . . . . 9 (𝜑𝑊 ∈ Mnd)
21 eqid 2620 . . . . . . . . . 10 (0g𝑊) = (0g𝑊)
229, 21pws0g 17307 . . . . . . . . 9 ((𝑊 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × {(0g𝑊)}) = (0g𝑋))
2320, 2, 22sylancl 693 . . . . . . . 8 (𝜑 → (𝐴 × {(0g𝑊)}) = (0g𝑋))
2423eqcomd 2626 . . . . . . 7 (𝜑 → (0g𝑋) = (𝐴 × {(0g𝑊)}))
2524eqeq2d 2630 . . . . . 6 (𝜑 → ((𝑥𝐴) = (0g𝑋) ↔ (𝑥𝐴) = (𝐴 × {(0g𝑊)})))
2625rabbidv 3184 . . . . 5 (𝜑 → {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (0g𝑋)} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
2717, 26syl5eq 2666 . . . 4 (𝜑 → ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
2827oveq2d 6651 . . 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 2620 . . . . . . 7 {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}
32 eqid 2620 . . . . . . 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 2620 . . . . . . 7 (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) = (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
358, 10, 21, 31, 32, 9, 33, 34pwssplit4 37478 . . . . . 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 1324 . . . . 5 (𝜑 → (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵)) ∈ ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) LMIso 𝑌))
37 brlmici 19050 . . . . 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 37477 . . . . 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 247 . . 3 (𝜑 → (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ∈ LNoeM)
4128, 40eqeltrd 2699 . 2 (𝜑 → (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) ∈ LNoeM)
428, 9, 10, 11, 12pwssplit1 19040 . . . . . . 7 ((𝑊 ∈ Mnd ∧ (𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
4320, 5, 7, 42syl3anc 1324 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
44 forn 6105 . . . . . 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 6651 . . . 4 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = (𝑋s (Base‘𝑋)))
47 pwslnmlem2.xn . . . . 5 (𝜑𝑋 ∈ LNoeM)
4811ressid 15916 . . . . 5 (𝑋 ∈ LNoeM → (𝑋s (Base‘𝑋)) = 𝑋)
4947, 48syl 17 . . . 4 (𝜑 → (𝑋s (Base‘𝑋)) = 𝑋)
5046, 49eqtrd 2654 . . 3 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = 𝑋)
5150, 47eqeltrd 2699 . 2 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) ∈ LNoeM)
52 eqid 2620 . . 3 (0g𝑋) = (0g𝑋)
53 eqid 2620 . . 3 ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})
54 eqid 2620 . . 3 (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) = (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}))
55 eqid 2620 . . 3 (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)))
5652, 53, 54, 55lmhmlnmsplit 37476 . 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 1324 1 (𝜑𝑍 ∈ LNoeM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1481  wcel 1988  {crab 2913  Vcvv 3195  cun 3565  cin 3566  wss 3567  c0 3907  {csn 4168   class class class wbr 4644  cmpt 4720   × cxp 5102  ccnv 5103  ran crn 5105  cres 5106  cima 5107  ontowfo 5874  cfv 5876  (class class class)co 6635  Basecbs 15838  s cress 15839  0gc0g 16081  s cpws 16088  Mndcmnd 17275  Grpcgrp 17403  LModclmod 18844   LMHom clmhm 19000   LMIso clmim 19001  𝑚 clmic 19002  LNoeMclnm 37464
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-rmo 2917  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-of 6882  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-pws 16091  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-submnd 17317  df-grp 17406  df-minusg 17407  df-sbg 17408  df-subg 17572  df-ghm 17639  df-cntz 17731  df-lsm 18032  df-cmn 18176  df-abl 18177  df-mgp 18471  df-ur 18483  df-ring 18530  df-lmod 18846  df-lss 18914  df-lsp 18953  df-lmhm 19003  df-lmim 19004  df-lmic 19005  df-lfig 37457  df-lnm 37465
This theorem is referenced by:  pwslnm  37483
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