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Theorem pwslnmlem2 42656
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 7749 . . . 4 (𝐴𝐵) ∈ V
54a1i 11 . . 3 (𝜑 → (𝐴𝐵) ∈ V)
6 ssun1 4170 . . . 4 𝐴 ⊆ (𝐴𝐵)
76a1i 11 . . 3 (𝜑𝐴 ⊆ (𝐴𝐵))
8 pwslnmlem2.z . . . 4 𝑍 = (𝑊s (𝐴𝐵))
9 pwslnmlem2.x . . . 4 𝑋 = (𝑊s 𝐴)
10 eqid 2725 . . . 4 (Base‘𝑍) = (Base‘𝑍)
11 eqid 2725 . . . 4 (Base‘𝑋) = (Base‘𝑋)
12 eqid 2725 . . . 4 (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) = (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))
138, 9, 10, 11, 12pwssplit3 20958 . . 3 ((𝑊 ∈ LMod ∧ (𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) ∈ (𝑍 LMHom 𝑋))
141, 5, 7, 13syl3anc 1368 . 2 (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) ∈ (𝑍 LMHom 𝑋))
15 fvex 6909 . . . . . 6 (0g𝑋) ∈ V
1612mptiniseg 6245 . . . . . 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 20762 . . . . . . . . . 10 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
19 grpmnd 18905 . . . . . . . . . 10 (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
201, 18, 193syl 18 . . . . . . . . 9 (𝜑𝑊 ∈ Mnd)
21 eqid 2725 . . . . . . . . . 10 (0g𝑊) = (0g𝑊)
229, 21pws0g 18733 . . . . . . . . 9 ((𝑊 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × {(0g𝑊)}) = (0g𝑋))
2320, 2, 22sylancl 584 . . . . . . . 8 (𝜑 → (𝐴 × {(0g𝑊)}) = (0g𝑋))
2423eqcomd 2731 . . . . . . 7 (𝜑 → (0g𝑋) = (𝐴 × {(0g𝑊)}))
2524eqeq2d 2736 . . . . . 6 (𝜑 → ((𝑥𝐴) = (0g𝑋) ↔ (𝑥𝐴) = (𝐴 × {(0g𝑊)})))
2625rabbidv 3426 . . . . 5 (𝜑 → {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (0g𝑋)} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
2717, 26eqtrid 2777 . . . 4 (𝜑 → ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
2827oveq2d 7435 . . 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 2725 . . . . . . 7 {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}
32 eqid 2725 . . . . . . 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 2725 . . . . . . 7 (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) = (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})})
358, 10, 21, 31, 32, 9, 33, 34pwssplit4 42652 . . . . . 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 1368 . . . . 5 (𝜑 → (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})} ↦ (𝑦𝐵)) ∈ ((𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) LMIso 𝑌))
37 brlmici 20966 . . . . 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 42651 . . . . 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 256 . . 3 (𝜑 → (𝑍s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥𝐴) = (𝐴 × {(0g𝑊)})}) ∈ LNoeM)
4128, 40eqeltrd 2825 . 2 (𝜑 → (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) ∈ LNoeM)
428, 9, 10, 11, 12pwssplit1 20956 . . . . . . 7 ((𝑊 ∈ Mnd ∧ (𝐴𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
4320, 5, 7, 42syl3anc 1368 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
44 forn 6813 . . . . . 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 7435 . . . 4 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = (𝑋s (Base‘𝑋)))
47 pwslnmlem2.xn . . . . 5 (𝜑𝑋 ∈ LNoeM)
4811ressid 17228 . . . . 5 (𝑋 ∈ LNoeM → (𝑋s (Base‘𝑋)) = 𝑋)
4947, 48syl 17 . . . 4 (𝜑 → (𝑋s (Base‘𝑋)) = 𝑋)
5046, 49eqtrd 2765 . . 3 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = 𝑋)
5150, 47eqeltrd 2825 . 2 (𝜑 → (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) ∈ LNoeM)
52 eqid 2725 . . 3 (0g𝑋) = (0g𝑋)
53 eqid 2725 . . 3 ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}) = ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})
54 eqid 2725 . . 3 (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)})) = (𝑍s ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)) “ {(0g𝑋)}))
55 eqid 2725 . . 3 (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴))) = (𝑋s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥𝐴)))
5652, 53, 54, 55lmhmlnmsplit 42650 . 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 1368 1 (𝜑𝑍 ∈ LNoeM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  {crab 3418  Vcvv 3461  cun 3942  cin 3943  wss 3944  c0 4322  {csn 4630   class class class wbr 5149  cmpt 5232   × cxp 5676  ccnv 5677  ran crn 5679  cres 5680  cima 5681  ontowfo 6547  cfv 6549  (class class class)co 7419  Basecbs 17183  s cress 17212  0gc0g 17424  s cpws 17431  Mndcmnd 18697  Grpcgrp 18898  LModclmod 20755   LMHom clmhm 20916   LMIso clmim 20917  𝑚 clmic 20918  LNoeMclnm 42638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-fz 13520  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-sca 17252  df-vsca 17253  df-ip 17254  df-tset 17255  df-ple 17256  df-ds 17258  df-hom 17260  df-cco 17261  df-0g 17426  df-prds 17432  df-pws 17434  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18744  df-grp 18901  df-minusg 18902  df-sbg 18903  df-subg 19086  df-ghm 19176  df-cntz 19280  df-lsm 19603  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-ring 20187  df-lmod 20757  df-lss 20828  df-lsp 20868  df-lmhm 20919  df-lmim 20920  df-lmic 20921  df-lfig 42631  df-lnm 42639
This theorem is referenced by:  pwslnm  42657
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