Theorem pwslnmlem2 43631

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 7722	. . . 4 ⊢ (𝐴 ∪ 𝐵) ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
6		ssun1 4128	. . . 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 2761	. . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍)
11		eqid 2761	. . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋)
12		eqid 2761	. . . 4 ⊢ (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) = (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))
13	8, 9, 10, 11, 12	pwssplit3 21116	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ V ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) ∈ (𝑍 LMHom 𝑋))
14	1, 5, 7, 13	syl3anc 1389	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) ∈ (𝑍 LMHom 𝑋))
15		fvex 6875	. . . . . 6 ⊢ (0g‘𝑋) ∈ V
16	12	mptiniseg 6221	. . . . . 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 20922	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
19		grpmnd 18973	. . . . . . . . . 10 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Mnd)
20	1, 18, 19	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Mnd)
21		eqid 2761	. . . . . . . . . 10 ⊢ (0g‘𝑊) = (0g‘𝑊)
22	9, 21	pws0g 18798	. . . . . . . . 9 ⊢ ((𝑊 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × {(0g‘𝑊)}) = (0g‘𝑋))
23	20, 2, 22	sylancl 595	. . . . . . . 8 ⊢ (𝜑 → (𝐴 × {(0g‘𝑊)}) = (0g‘𝑋))
24	23	eqcomd 2767	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑋) = (𝐴 × {(0g‘𝑊)}))
25	24	eqeq2d 2772	. . . . . 6 ⊢ (𝜑 → ((𝑥 ↾ 𝐴) = (0g‘𝑋) ↔ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})))
26	25	rabbidv 3420	. . . . 5 ⊢ (𝜑 → {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (0g‘𝑋)} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})})
27	17, 26	eqtrid 2808	. . . 4 ⊢ (𝜑 → (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})})
28	27	oveq2d 7407	. . 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 2761	. . . . . . 7 ⊢ {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})} = {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}
32		eqid 2761	. . . . . . 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 2761	. . . . . . 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 43627	. . . . . 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 1389	. . . . 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 21124	. . . . 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 43626	. . . . 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 259	. . 3 ⊢ (𝜑 → (𝑍 ↾s {𝑥 ∈ (Base‘𝑍) ∣ (𝑥 ↾ 𝐴) = (𝐴 × {(0g‘𝑊)})}) ∈ LNoeM)
41	28, 40	eqeltrd 2861	. 2 ⊢ (𝜑 → (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) ∈ LNoeM)
42	8, 9, 10, 11, 12	pwssplit1 21114	. . . . . . 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 1389	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)):(Base‘𝑍)–onto→(Base‘𝑋))
44		forn 6776	. . . . . 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 7407	. . . 4 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = (𝑋 ↾s (Base‘𝑋)))
47		pwslnmlem2.xn	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ LNoeM)
48	11	ressid 17271	. . . . 5 ⊢ (𝑋 ∈ LNoeM → (𝑋 ↾s (Base‘𝑋)) = 𝑋)
49	47, 48	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 ↾s (Base‘𝑋)) = 𝑋)
50	46, 49	eqtrd 2796	. . 3 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = 𝑋)
51	50, 47	eqeltrd 2861	. 2 ⊢ (𝜑 → (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) ∈ LNoeM)
52		eqid 2761	. . 3 ⊢ (0g‘𝑋) = (0g‘𝑋)
53		eqid 2761	. . 3 ⊢ (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}) = (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})
54		eqid 2761	. . 3 ⊢ (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)})) = (𝑍 ↾s (◡(𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)) “ {(0g‘𝑋)}))
55		eqid 2761	. . 3 ⊢ (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴))) = (𝑋 ↾s ran (𝑥 ∈ (Base‘𝑍) ↦ (𝑥 ↾ 𝐴)))
56	52, 53, 54, 55	lmhmlnmsplit 43625	. 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 1389	1 ⊢ (𝜑 → 𝑍 ∈ LNoeM)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  {csn 4579   class class class wbr 5097   ↦ cmpt 5178   × cxp 5641  ^◡ccnv 5642  ran crn 5644   ↾ cres 5645   “ cima 5646  –onto→wfo 6514  ‘cfv 6516  (class class class)co 7391  Basecbs 17236   ↾_s cress 17257  0_gc0g 17459   ↑_s cpws 17466  Mndcmnd 18759  Grpcgrp 18966  LModclmod 20915   LMHom clmhm 21074   LMIso clmim 21075   ≃_𝑚 clmic 21076  LNoeMclnm 43613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-of 7655  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-er 8672  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9382  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12476  df-z 12563  df-dec 12683  df-uz 12834  df-fz 13507  df-struct 17174  df-sets 17191  df-slot 17209  df-ndx 17221  df-base 17237  df-ress 17258  df-plusg 17290  df-mulr 17291  df-sca 17293  df-vsca 17294  df-ip 17295  df-tset 17296  df-ple 17297  df-ds 17299  df-hom 17301  df-cco 17302  df-0g 17461  df-prds 17467  df-pws 17469  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-grp 18969  df-minusg 18970  df-sbg 18971  df-subg 19156  df-ghm 19245  df-cntz 19348  df-lsm 19667  df-cmn 19813  df-abl 19814  df-mgp 20178  df-rng 20190  df-ur 20219  df-ring 20272  df-lmod 20917  df-lss 20987  df-lsp 21027  df-lmhm 21077  df-lmim 21078  df-lmic 21079  df-lfig 43606  df-lnm 43614

This theorem is referenced by:  pwslnm  43632