Theorem pwslnmlem2 40037

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 = (𝑊 ↑s 𝐴)
pwslnmlem2.y	⊢ 𝑌 = (𝑊 ↑s 𝐵)
pwslnmlem2.z	⊢ 𝑍 = (𝑊 ↑s (𝐴 ∪ 𝐵))
pwslnmlem2.w	⊢ (𝜑 → 𝑊 ∈ LMod)
pwslnmlem2.dj	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
pwslnmlem2.xn	⊢ (𝜑 → 𝑋 ∈ LNoeM)
pwslnmlem2.yn	⊢ (𝜑 → 𝑌 ∈ LNoeM)

Assertion

Ref	Expression
pwslnmlem2	⊢ (𝜑 → 𝑍 ∈ LNoeM)

Proof of Theorem pwslnmlem2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwslnmlem2.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ LMod)
2		pwslnmlem2.a	. . . . 5 ⊢ 𝐴 ∈ V
3		pwslnmlem2.b	. . . . 5 ⊢ 𝐵 ∈ V
4	2, 3	unex 7449	. . . 4 ⊢ (𝐴 ∪ 𝐵) ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
6		ssun1 4099	. . . 4 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
7	6	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (𝐴 ∪ 𝐵))
8		pwslnmlem2.z	. . . 4 ⊢ 𝑍 = (𝑊 ↑s (𝐴 ∪ 𝐵))
9		pwslnmlem2.x	. . . 4 ⊢ 𝑋 = (𝑊 ↑s 𝐴)
10		eqid 2798	. . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍)
11		eqid 2798	. . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋)
12		eqid 2798	. . . 4 ⊢ (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) = (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))
13	8, 9, 10, 11, 12	pwssplit3 19826	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) ∈ (𝑍 LMHom 𝑋))
14	1, 5, 7, 13	syl3anc 1368	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) ∈ (𝑍 LMHom 𝑋))
15		fvex 6658	. . . . . 6 ⊢ (0g‘𝑋) ∈ V
16	12	mptiniseg 6060	. . . . . 6 ⊢ ((0g‘𝑋) ∈ V → (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (0g‘𝑋)})
17	15, 16	ax-mp 5	. . . . 5 ⊢ (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (0g‘𝑋)}
18		lmodgrp 19634	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
19		grpmnd 18102	. . . . . . . . . 10 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
20	1, 18, 19	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Mnd)
21		eqid 2798	. . . . . . . . . 10 ⊢ (0g‘𝑊) = (0g‘𝑊)
22	9, 21	pws0g 17939	. . . . . . . . 9 ⊢ ((𝑊 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × {(0g‘𝑊)}) = (0g‘𝑋))
23	20, 2, 22	sylancl 589	. . . . . . . 8 ⊢ (𝜑 → (𝐴 × {(0g‘𝑊)}) = (0g‘𝑋))
24	23	eqcomd 2804	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑋) = (𝐴 × {(0g‘𝑊)}))
25	24	eqeq2d 2809	. . . . . 6 ⊢ (𝜑 → ((𝑥 ↾ 𝐴) = (0g‘𝑋) ↔ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})))
26	25	rabbidv 3427	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (0g‘𝑋)} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})})
27	17, 26	syl5eq 2845	. . . 4 ⊢ (𝜑 → (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})})
28	27	oveq2d 7151	. . 3 ⊢ (𝜑 → (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) = (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}))
29		pwslnmlem2.yn	. . . 4 ⊢ (𝜑 → 𝑌 ∈ LNoeM)
30		pwslnmlem2.dj	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
31		eqid 2798	. . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}
32		eqid 2798	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} ↦ (𝑦 ↾ 𝐵)) = (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} ↦ (𝑦 ↾ 𝐵))
33		pwslnmlem2.y	. . . . . . 7 ⊢ 𝑌 = (𝑊 ↑s 𝐵)
34		eqid 2798	. . . . . . 7 ⊢ (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) = (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})})
35	8, 10, 21, 31, 32, 9, 33, 34	pwssplit4 40033	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ V ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} ↦ (𝑦 ↾ 𝐵)) ∈ ((𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) LMIso 𝑌))
36	1, 5, 30, 35	syl3anc 1368	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} ↦ (𝑦 ↾ 𝐵)) ∈ ((𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) LMIso 𝑌))
37		brlmici 19834	. . . . 5 ⊢ ((𝑦 ∈ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} ↦ (𝑦 ↾ 𝐵)) ∈ ((𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) LMIso 𝑌) → (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ≃𝑚 𝑌)
38		lnmlmic 40032	. . . . 5 ⊢ ((𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ≃𝑚 𝑌 → ((𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ∈ LNoeM ↔ 𝑌 ∈ LNoeM))
39	36, 37, 38	3syl 18	. . . 4 ⊢ (𝜑 → ((𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ∈ LNoeM ↔ 𝑌 ∈ LNoeM))
40	29, 39	mpbird 260	. . 3 ⊢ (𝜑 → (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ∈ LNoeM)
41	28, 40	eqeltrd 2890	. 2 ⊢ (𝜑 → (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) ∈ LNoeM)
42	8, 9, 10, 11, 12	pwssplit1 19824	. . . . . . 7 ⊢ ((𝑊 ∈ Mnd ∧ (𝐴 ∪ 𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
43	20, 5, 7, 42	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
44		forn 6568	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)):(Base‘𝑍)–onto→(Base‘𝑋) → ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) = (Base‘𝑋))
45	43, 44	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) = (Base‘𝑋))
46	45	oveq2d 7151	. . . 4 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = (𝑋 ↾s (Base‘𝑋)))
47		pwslnmlem2.xn	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ LNoeM)
48	11	ressid 16551	. . . . 5 ⊢ (𝑋 ∈ LNoeM → (𝑋 ↾s (Base‘𝑋)) = 𝑋)
49	47, 48	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 ↾s (Base‘𝑋)) = 𝑋)
50	46, 49	eqtrd 2833	. . 3 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = 𝑋)
51	50, 47	eqeltrd 2890	. 2 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) ∈ LNoeM)
52		eqid 2798	. . 3 ⊢ (0g‘𝑋) = (0g‘𝑋)
53		eqid 2798	. . 3 ⊢ (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})
54		eqid 2798	. . 3 ⊢ (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) = (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}))
55		eqid 2798	. . 3 ⊢ (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)))
56	52, 53, 54, 55	lmhmlnmsplit 40031	. 2 ⊢ (((𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) ∈ (𝑍 LMHom 𝑋) ∧ (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) ∈ LNoeM ∧ (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) ∈ LNoeM) → 𝑍 ∈ LNoeM)
57	14, 41, 51, 56	syl3anc 1368	1 ⊢ (𝜑 → 𝑍 ∈ LNoeM)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  {csn 4525   class class class wbr 5030   ↦ cmpt 5110   × cxp 5517  ^◡ccnv 5518  ran crn 5520   ↾ cres 5521   “ cima 5522  –onto→wfo 6322  ‘cfv 6324  (class class class)co 7135  Basecbs 16475   ↾_s cress 16476  0_gc0g 16705   ↑_s cpws 16712  Mndcmnd 17903  Grpcgrp 18095  LModclmod 19627   LMHom clmhm 19784   LMIso clmim 19785   ≃_𝑚 clmic 19786  LNoeMclnm 40019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-hom 16581  df-cco 16582  df-0g 16707  df-prds 16713  df-pws 16715  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-subg 18268  df-ghm 18348  df-cntz 18439  df-lsm 18753  df-cmn 18900  df-abl 18901  df-mgp 19233  df-ur 19245  df-ring 19292  df-lmod 19629  df-lss 19697  df-lsp 19737  df-lmhm 19787  df-lmim 19788  df-lmic 19789  df-lfig 40012  df-lnm 40020

This theorem is referenced by:  pwslnm  40038