Theorem fincmpzer 14711

Description: Finite composite of identity elements.

Hypothesis

Ref	Expression
fincmpzer.1	|- U = (Id` G)

Assertion

Ref	Expression
fincmpzer	|- ((N e. (ZZ>=` M) /\ G e. (Magma i^i ExId )) -> prod_k e. (M...N)GU = U)

Distinct variable groups:   k,M   k,N   U,k

Proof of Theorem fincmpzer

Step	Hyp	Ref	Expression
1		fincmpzer.1	. 2 |- U = (Id` G)
2		eleq1 1957	. . . . . 6 |- (g = G -> (g e. (Magma i^i ExId ) <-> G e. (Magma i^i ExId )))
3	2	anbi2d 678	. . . . 5 |- (g = G -> ((N e. (ZZ>=` M) /\ g e. (Magma i^i ExId )) <-> (N e. (ZZ>=` M) /\ G e. (Magma i^i ExId ))))
4		fveq2 4681	. . . . . . 7 |- (g = G -> (Id` g) = (Id` G))
5	4	eqeq2d 1895	. . . . . 6 |- (g = G -> (U = (Id` g) <-> U = (Id` G)))
6		prodeq3 14663	. . . . . . 7 |- (g = G -> prod_k e. (M...N)gU = prod_k e. (M...N)GU)
7	6	eqeq1d 1892	. . . . . 6 |- (g = G -> (prod_k e. (M...N)gU = U <-> prod_k e. (M...N)GU = U))
8	5, 7	imbi12d 688	. . . . 5 |- (g = G -> ((U = (Id` g) -> prod_k e. (M...N)gU = U) <-> (U = (Id` G) -> prod_k e. (M...N)GU = U)))
9	3, 8	imbi12d 688	. . . 4 |- (g = G -> (((N e. (ZZ>=` M) /\ g e. (Magma i^i ExId )) -> (U = (Id` g) -> prod_k e. (M...N)gU = U)) <-> ((N e. (ZZ>=` M) /\ G e. (Magma i^i ExId )) -> (U = (Id` G) -> prod_k e. (M...N)GU = U))))
10		ax-17 1317	. . . . . . 7 |- (U = (Id` g) -> A.k U = (Id` g))
11		alral 2153	. . . . . . . 8 |- (A.k U = (Id` g) -> A.k e. (M...N)U = (Id` g))
12		visset 2295	. . . . . . . 8 |- g e. _V
13	11, 12	prodeq3d 14668	. . . . . . 7 |- (A.k U = (Id` g) -> prod_k e. (M...N)gU = prod_k e. (M...N)g(Id` g))
14	10, 13	syl 12	. . . . . 6 |- (U = (Id` g) -> prod_k e. (M...N)gU = prod_k e. (M...N)g(Id` g))
15		id 73	. . . . . 6 |- (U = (Id` g) -> U = (Id` g))
16	14, 15	eqeq12d 1899	. . . . 5 |- (U = (Id` g) -> (prod_k e. (M...N)gU = U <-> prod_k e. (M...N)g(Id` g) = (Id` g)))
17		opreq2 4890	. . . . . . . . . 10 |- (j = M -> (M...j) = (M...M))
18	17	prodeq1d 14666	. . . . . . . . 9 |- (j = M -> prod_k e. (M...j)g(Id` g) = prod_k e. (M...M)g(Id` g))
19	18	eqeq1d 1892	. . . . . . . 8 |- (j = M -> (prod_k e. (M...j)g(Id` g) = (Id` g) <-> prod_k e. (M...M)g(Id` g) = (Id` g)))
20	19	imbi2d 674	. . . . . . 7 |- (j = M -> ((g e. (Magma i^i ExId ) -> prod_k e. (M...j)g(Id` g) = (Id` g)) <-> (g e. (Magma i^i ExId ) -> prod_k e. (M...M)g(Id` g) = (Id` g))))
21		opreq2 4890	. . . . . . . . . 10 |- (j = l -> (M...j) = (M...l))
22	21	prodeq1d 14666	. . . . . . . . 9 |- (j = l -> prod_k e. (M...j)g(Id` g) = prod_k e. (M...l)g(Id` g))
23	22	eqeq1d 1892	. . . . . . . 8 |- (j = l -> (prod_k e. (M...j)g(Id` g) = (Id` g) <-> prod_k e. (M...l)g(Id` g) = (Id` g)))
24	23	imbi2d 674	. . . . . . 7 |- (j = l -> ((g e. (Magma i^i ExId ) -> prod_k e. (M...j)g(Id` g) = (Id` g)) <-> (g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g))))
25		opreq2 4890	. . . . . . . . . 10 |- (j = (l + 1) -> (M...j) = (M...(l + 1)))
26	25	prodeq1d 14666	. . . . . . . . 9 |- (j = (l + 1) -> prod_k e. (M...j)g(Id` g) = prod_k e. (M...(l + 1))g(Id` g))
27	26	eqeq1d 1892	. . . . . . . 8 |- (j = (l + 1) -> (prod_k e. (M...j)g(Id` g) = (Id` g) <-> prod_k e. (M...(l + 1))g(Id` g) = (Id` g)))
28	27	imbi2d 674	. . . . . . 7 |- (j = (l + 1) -> ((g e. (Magma i^i ExId ) -> prod_k e. (M...j)g(Id` g) = (Id` g)) <-> (g e. (Magma i^i ExId ) -> prod_k e. (M...(l + 1))g(Id` g) = (Id` g))))
29		opreq2 4890	. . . . . . . . . 10 |- (j = N -> (M...j) = (M...N))
30	29	prodeq1d 14666	. . . . . . . . 9 |- (j = N -> prod_k e. (M...j)g(Id` g) = prod_k e. (M...N)g(Id` g))
31	30	eqeq1d 1892	. . . . . . . 8 |- (j = N -> (prod_k e. (M...j)g(Id` g) = (Id` g) <-> prod_k e. (M...N)g(Id` g) = (Id` g)))
32	31	imbi2d 674	. . . . . . 7 |- (j = N -> ((g e. (Magma i^i ExId ) -> prod_k e. (M...j)g(Id` g) = (Id` g)) <-> (g e. (Magma i^i ExId ) -> prod_k e. (M...N)g(Id` g) = (Id` g))))
33		fvex 4689	. . . . . . . . 9 |- (Id` g) e. _V
34		eqidd 1885	. . . . . . . . . 10 |- (k = M -> (Id` g) = (Id` g))
35	34, 12	fprod1i 14673	. . . . . . . . 9 |- (((Id` g) e. _V /\ M e. ZZ) -> prod_k e. (M...M)g(Id` g) = (Id` g))
36	33, 35	mpan 759	. . . . . . . 8 |- (M e. ZZ -> prod_k e. (M...M)g(Id` g) = (Id` g))
37	36	a1d 15	. . . . . . 7 |- (M e. ZZ -> (g e. (Magma i^i ExId ) -> prod_k e. (M...M)g(Id` g) = (Id` g)))
38	12	fprodp1s 14682	. . . . . . . . . 10 |- ((l e. (ZZ>=` M) /\ A.k e. (M...(l + 1))(Id` g) e. _V) -> prod_k e. (M...(l + 1))g(Id` g) = (prod_k e. (M...l)g(Id` g)g[_(l + 1) / k]_(Id` g)))
39		simp1 876	. . . . . . . . . 10 |- ((l e. (ZZ>=` M) /\ (g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g)) /\ g e. (Magma i^i ExId )) -> l e. (ZZ>=` M))
40	33	a1i 8	. . . . . . . . . . 11 |- (k e. (M...(l + 1)) -> (Id` g) e. _V)
41	40	rgen 2159	. . . . . . . . . 10 |- A.k e. (M...(l + 1))(Id` g) e. _V
42	38, 39, 41	sylancl 525	. . . . . . . . 9 |- ((l e. (ZZ>=` M) /\ (g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g)) /\ g e. (Magma i^i ExId )) -> prod_k e. (M...(l + 1))g(Id` g) = (prod_k e. (M...l)g(Id` g)g[_(l + 1) / k]_(Id` g)))
43		idd 75	. . . . . . . . . . 11 |- (l e. (ZZ>=` M) -> ((g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g)) -> (g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g))))
44	43	3imp 1061	. . . . . . . . . 10 |- ((l e. (ZZ>=` M) /\ (g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g)) /\ g e. (Magma i^i ExId )) -> prod_k e. (M...l)g(Id` g) = (Id` g))
45		oprex 4907	. . . . . . . . . . . 12 |- (l + 1) e. _V
46		ax-17 1317	. . . . . . . . . . . . 13 |- (y e. (Id` g) -> A.k y e. (Id` g))
47	46	csbconstgf 2551	. . . . . . . . . . . 12 |- ((l + 1) e. _V -> [_(l + 1) / k]_(Id` g) = (Id` g))
48	45, 47	ax-mp 7	. . . . . . . . . . 11 |- [_(l + 1) / k]_(Id` g) = (Id` g)
49	48	a1i 8	. . . . . . . . . 10 |- ((l e. (ZZ>=` M) /\ (g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g)) /\ g e. (Magma i^i ExId )) -> [_(l + 1) / k]_(Id` g) = (Id` g))
50	44, 49	opreq12d 4900	. . . . . . . . 9 |- ((l e. (ZZ>=` M) /\ (g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g)) /\ g e. (Magma i^i ExId )) -> (prod_k e. (M...l)g(Id` g)g[_(l + 1) / k]_(Id` g)) = ((Id` g)g(Id` g)))
51		simp3 878	. . . . . . . . . . 11 |- ((l e. (ZZ>=` M) /\ (g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g)) /\ g e. (Magma i^i ExId )) -> g e. (Magma i^i ExId ))
52		eqid 1884	. . . . . . . . . . . . 13 |- ran g = ran g
53		eqid 1884	. . . . . . . . . . . . 13 |- (Id` g) = (Id` g)
54	52, 53	iorlid 10375	. . . . . . . . . . . 12 |- (g e. (Magma i^i ExId ) -> (Id` g) e. ran g)
55	54	3ad2ant3 899	. . . . . . . . . . 11 |- ((l e. (ZZ>=` M) /\ (g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g)) /\ g e. (Magma i^i ExId )) -> (Id` g) e. ran g)
56	52, 53	cmpidelt 10376	. . . . . . . . . . 11 |- ((g e. (Magma i^i ExId ) /\ (Id` g) e. ran g) -> (((Id` g)g(Id` g)) = (Id` g) /\ ((Id` g)g(Id` g)) = (Id` g)))
57	51, 55, 56	syl11anc 524	. . . . . . . . . 10 |- ((l e. (ZZ>=` M) /\ (g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g)) /\ g e. (Magma i^i ExId )) -> (((Id` g)g(Id` g)) = (Id` g) /\ ((Id` g)g(Id` g)) = (Id` g)))
58	57	simplld 348	. . . . . . . . 9 |- ((l e. (ZZ>=` M) /\ (g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g)) /\ g e. (Magma i^i ExId )) -> ((Id` g)g(Id` g)) = (Id` g))
59	42, 50, 58	3eqtrd 1929	. . . . . . . 8 |- ((l e. (ZZ>=` M) /\ (g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g)) /\ g e. (Magma i^i ExId )) -> prod_k e. (M...(l + 1))g(Id` g) = (Id` g))
60	59	3exp 1066	. . . . . . 7 |- (l e. (ZZ>=` M) -> ((g e. (Magma i^i ExId ) -> prod_k e. (M...l)g(Id` g) = (Id` g)) -> (g e. (Magma i^i ExId ) -> prod_k e. (M...(l + 1))g(Id` g) = (Id` g))))
61	20, 24, 28, 32, 37, 60	uzind4 7619	. . . . . 6 |- (N e. (ZZ>=` M) -> (g e. (Magma i^i ExId ) -> prod_k e. (M...N)g(Id` g) = (Id` g)))
62	61	imp 377	. . . . 5 |- ((N e. (ZZ>=` M) /\ g e. (Magma i^i ExId )) -> prod_k e. (M...N)g(Id` g) = (Id` g))
63	16, 62	syl5cbir 228	. . . 4 |- ((N e. (ZZ>=` M) /\ g e. (Magma i^i ExId )) -> (U = (Id` g) -> prod_k e. (M...N)gU = U))
64	9, 63	vtoclg 2346	. . 3 |- (G e. (Magma i^i ExId ) -> ((N e. (ZZ>=` M) /\ G e. (Magma i^i ExId )) -> (U = (Id` G) -> prod_k e. (M...N)GU = U)))
65	64	anabsi7 555	. 2 |- ((N e. (ZZ>=` M) /\ G e. (Magma i^i ExId )) -> (U = (Id` G) -> prod_k e. (M...N)GU = U))
66	1, 65	mpi 55	1 |- ((N e. (ZZ>=` M) /\ G e. (Magma i^i ExId )) -> prod_k e. (M...N)GU = U)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  [_csb 2540   i^i cin 2592  ran crn 3987  ` cfv 3998  (class class class)co 4884  1c1 6387   + caddc 6389  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637  Idcgi 9312   ExId cexid 10361  Magmacmagm 10365  prod_cprd2 14654

This theorem is referenced by:  svli2 14826

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-gid 9317  df-exid 10362  df-mgm 10366  df-prod 14653  df-prod2 14655

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