Theorem pwslnmlem2 43331

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 7689	. . . 4 ⊢ (𝐴 ∪ 𝐵) ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
6		ssun1 4130	. . . 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 2736	. . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍)
11		eqid 2736	. . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋)
12		eqid 2736	. . . 4 ⊢ (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) = (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))
13	8, 9, 10, 11, 12	pwssplit3 21013	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) ∈ (𝑍 LMHom 𝑋))
14	1, 5, 7, 13	syl3anc 1373	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) ∈ (𝑍 LMHom 𝑋))
15		fvex 6847	. . . . . 6 ⊢ (0g‘𝑋) ∈ V
16	12	mptiniseg 6197	. . . . . 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 20818	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
19		grpmnd 18870	. . . . . . . . . 10 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
20	1, 18, 19	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Mnd)
21		eqid 2736	. . . . . . . . . 10 ⊢ (0g‘𝑊) = (0g‘𝑊)
22	9, 21	pws0g 18698	. . . . . . . . 9 ⊢ ((𝑊 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × {(0g‘𝑊)}) = (0g‘𝑋))
23	20, 2, 22	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → (𝐴 × {(0g‘𝑊)}) = (0g‘𝑋))
24	23	eqcomd 2742	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑋) = (𝐴 × {(0g‘𝑊)}))
25	24	eqeq2d 2747	. . . . . 6 ⊢ (𝜑 → ((𝑥 ↾ 𝐴) = (0g‘𝑋) ↔ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})))
26	25	rabbidv 3406	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (0g‘𝑋)} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})})
27	17, 26	eqtrid 2783	. . . 4 ⊢ (𝜑 → (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})})
28	27	oveq2d 7374	. . 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 2736	. . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}
32		eqid 2736	. . . . . . 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 2736	. . . . . . 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 43327	. . . . . 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 1373	. . . . 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 21021	. . . . 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 43326	. . . . 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 257	. . 3 ⊢ (𝜑 → (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ∈ LNoeM)
41	28, 40	eqeltrd 2836	. 2 ⊢ (𝜑 → (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) ∈ LNoeM)
42	8, 9, 10, 11, 12	pwssplit1 21011	. . . . . . 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 1373	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
44		forn 6749	. . . . . 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 7374	. . . 4 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = (𝑋 ↾s (Base‘𝑋)))
47		pwslnmlem2.xn	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ LNoeM)
48	11	ressid 17171	. . . . 5 ⊢ (𝑋 ∈ LNoeM → (𝑋 ↾s (Base‘𝑋)) = 𝑋)
49	47, 48	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 ↾s (Base‘𝑋)) = 𝑋)
50	46, 49	eqtrd 2771	. . 3 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = 𝑋)
51	50, 47	eqeltrd 2836	. 2 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) ∈ LNoeM)
52		eqid 2736	. . 3 ⊢ (0g‘𝑋) = (0g‘𝑋)
53		eqid 2736	. . 3 ⊢ (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})
54		eqid 2736	. . 3 ⊢ (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) = (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}))
55		eqid 2736	. . 3 ⊢ (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)))
56	52, 53, 54, 55	lmhmlnmsplit 43325	. 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 1373	1 ⊢ (𝜑 → 𝑍 ∈ LNoeM)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   ∪ cun 3899   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  {csn 4580   class class class wbr 5098   ↦ cmpt 5179   × cxp 5622  ^◡ccnv 5623  ran crn 5625   ↾ cres 5626   “ cima 5627  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7358  Basecbs 17136   ↾_s cress 17157  0_gc0g 17359   ↑_s cpws 17366  Mndcmnd 18659  Grpcgrp 18863  LModclmod 20811   LMHom clmhm 20971   LMIso clmim 20972   ≃_𝑚 clmic 20973  LNoeMclnm 43313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-hom 17201  df-cco 17202  df-0g 17361  df-prds 17367  df-pws 17369  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19053  df-ghm 19142  df-cntz 19246  df-lsm 19565  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-ring 20170  df-lmod 20813  df-lss 20883  df-lsp 20923  df-lmhm 20974  df-lmim 20975  df-lmic 20976  df-lfig 43306  df-lnm 43314

This theorem is referenced by:  pwslnm  43332