Theorem clfsebs 14707

Description: Closure of a finite composite of elements of the base set of an internal operation.

Hypothesis

Ref	Expression
clfsebs.1	|- X = dom dom G

Assertion

Ref	Expression
clfsebs	|- ((N e. (ZZ>=` M) /\ G e. Magma /\ A.k e. (M...N)A e. X) -> prod_k e. (M...N)GA e. X)

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

Proof of Theorem clfsebs

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . . . 7 |- (g = G -> (g e. Magma <-> G e. Magma))
2		dmeq 4157	. . . . . . . . . 10 |- (g = G -> dom g = dom G)
3	2	dmeqd 4159	. . . . . . . . 9 |- (g = G -> dom dom g = dom dom G)
4	3	eleq2d 1964	. . . . . . . 8 |- (g = G -> (A e. dom dom g <-> A e. dom dom G))
5	4	ralbidv 2123	. . . . . . 7 |- (g = G -> (A.k e. (M...N)A e. dom dom g <-> A.k e. (M...N)A e. dom dom G))
6	1, 5	3anbi23d 1171	. . . . . 6 |- (g = G -> ((N e. (ZZ>=` M) /\ g e. Magma /\ A.k e. (M...N)A e. dom dom g) <-> (N e. (ZZ>=` M) /\ G e. Magma /\ A.k e. (M...N)A e. dom dom G)))
7		prodeq3 14663	. . . . . . 7 |- (g = G -> prod_k e. (M...N)gA = prod_k e. (M...N)GA)
8	7, 3	eleq12d 1965	. . . . . 6 |- (g = G -> (prod_k e. (M...N)gA e. dom dom g <-> prod_k e. (M...N)GA e. dom dom G))
9	6, 8	imbi12d 688	. . . . 5 |- (g = G -> (((N e. (ZZ>=` M) /\ g e. Magma /\ A.k e. (M...N)A e. dom dom g) -> prod_k e. (M...N)gA e. dom dom g) <-> ((N e. (ZZ>=` M) /\ G e. Magma /\ A.k e. (M...N)A e. dom dom G) -> prod_k e. (M...N)GA e. dom dom G)))
10		opreq2 4890	. . . . . . . . . 10 |- (j = M -> (M...j) = (M...M))
11	10	raleqdv 2269	. . . . . . . . 9 |- (j = M -> (A.k e. (M...j)A e. dom dom g <-> A.k e. (M...M)A e. dom dom g))
12	10	prodeq1d 14666	. . . . . . . . . 10 |- (j = M -> prod_k e. (M...j)gA = prod_k e. (M...M)gA)
13	12	eleq1d 1963	. . . . . . . . 9 |- (j = M -> (prod_k e. (M...j)gA e. dom dom g <-> prod_k e. (M...M)gA e. dom dom g))
14	11, 13	imbi12d 688	. . . . . . . 8 |- (j = M -> ((A.k e. (M...j)A e. dom dom g -> prod_k e. (M...j)gA e. dom dom g) <-> (A.k e. (M...M)A e. dom dom g -> prod_k e. (M...M)gA e. dom dom g)))
15	14	imbi2d 674	. . . . . . 7 |- (j = M -> ((g e. Magma -> (A.k e. (M...j)A e. dom dom g -> prod_k e. (M...j)gA e. dom dom g)) <-> (g e. Magma -> (A.k e. (M...M)A e. dom dom g -> prod_k e. (M...M)gA e. dom dom g))))
16		opreq2 4890	. . . . . . . . . 10 |- (j = m -> (M...j) = (M...m))
17	16	raleqdv 2269	. . . . . . . . 9 |- (j = m -> (A.k e. (M...j)A e. dom dom g <-> A.k e. (M...m)A e. dom dom g))
18	16	prodeq1d 14666	. . . . . . . . . 10 |- (j = m -> prod_k e. (M...j)gA = prod_k e. (M...m)gA)
19	18	eleq1d 1963	. . . . . . . . 9 |- (j = m -> (prod_k e. (M...j)gA e. dom dom g <-> prod_k e. (M...m)gA e. dom dom g))
20	17, 19	imbi12d 688	. . . . . . . 8 |- (j = m -> ((A.k e. (M...j)A e. dom dom g -> prod_k e. (M...j)gA e. dom dom g) <-> (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)))
21	20	imbi2d 674	. . . . . . 7 |- (j = m -> ((g e. Magma -> (A.k e. (M...j)A e. dom dom g -> prod_k e. (M...j)gA e. dom dom g)) <-> (g e. Magma -> (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g))))
22		opreq2 4890	. . . . . . . . . 10 |- (j = (m + 1) -> (M...j) = (M...(m + 1)))
23	22	raleqdv 2269	. . . . . . . . 9 |- (j = (m + 1) -> (A.k e. (M...j)A e. dom dom g <-> A.k e. (M...(m + 1))A e. dom dom g))
24	22	prodeq1d 14666	. . . . . . . . . 10 |- (j = (m + 1) -> prod_k e. (M...j)gA = prod_k e. (M...(m + 1))gA)
25	24	eleq1d 1963	. . . . . . . . 9 |- (j = (m + 1) -> (prod_k e. (M...j)gA e. dom dom g <-> prod_k e. (M...(m + 1))gA e. dom dom g))
26	23, 25	imbi12d 688	. . . . . . . 8 |- (j = (m + 1) -> ((A.k e. (M...j)A e. dom dom g -> prod_k e. (M...j)gA e. dom dom g) <-> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...(m + 1))gA e. dom dom g)))
27	26	imbi2d 674	. . . . . . 7 |- (j = (m + 1) -> ((g e. Magma -> (A.k e. (M...j)A e. dom dom g -> prod_k e. (M...j)gA e. dom dom g)) <-> (g e. Magma -> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...(m + 1))gA e. dom dom g))))
28		opreq2 4890	. . . . . . . . . 10 |- (j = N -> (M...j) = (M...N))
29	28	raleqdv 2269	. . . . . . . . 9 |- (j = N -> (A.k e. (M...j)A e. dom dom g <-> A.k e. (M...N)A e. dom dom g))
30	28	prodeq1d 14666	. . . . . . . . . 10 |- (j = N -> prod_k e. (M...j)gA = prod_k e. (M...N)gA)
31	30	eleq1d 1963	. . . . . . . . 9 |- (j = N -> (prod_k e. (M...j)gA e. dom dom g <-> prod_k e. (M...N)gA e. dom dom g))
32	29, 31	imbi12d 688	. . . . . . . 8 |- (j = N -> ((A.k e. (M...j)A e. dom dom g -> prod_k e. (M...j)gA e. dom dom g) <-> (A.k e. (M...N)A e. dom dom g -> prod_k e. (M...N)gA e. dom dom g)))
33	32	imbi2d 674	. . . . . . 7 |- (j = N -> ((g e. Magma -> (A.k e. (M...j)A e. dom dom g -> prod_k e. (M...j)gA e. dom dom g)) <-> (g e. Magma -> (A.k e. (M...N)A e. dom dom g -> prod_k e. (M...N)gA e. dom dom g))))
34		fprod1s1 14679	. . . . . . . . . 10 |- ((g e. Magma /\ M e. ZZ /\ A.k e. (M...M)A e. dom dom g) -> prod_k e. (M...M)gA = [_M / k]_A)
35		ra4csbela 2587	. . . . . . . . . . . 12 |- ((M e. (M...M) /\ A.k e. (M...M)A e. dom dom g) -> [_M / k]_A e. dom dom g)
36		elfz3 7661	. . . . . . . . . . . 12 |- (M e. ZZ -> M e. (M...M))
37	35, 36	sylan 497	. . . . . . . . . . 11 |- ((M e. ZZ /\ A.k e. (M...M)A e. dom dom g) -> [_M / k]_A e. dom dom g)
38	37	3adant1 894	. . . . . . . . . 10 |- ((g e. Magma /\ M e. ZZ /\ A.k e. (M...M)A e. dom dom g) -> [_M / k]_A e. dom dom g)
39	34, 38	eqeltrd 1971	. . . . . . . . 9 |- ((g e. Magma /\ M e. ZZ /\ A.k e. (M...M)A e. dom dom g) -> prod_k e. (M...M)gA e. dom dom g)
40	39	3exp 1066	. . . . . . . 8 |- (g e. Magma -> (M e. ZZ -> (A.k e. (M...M)A e. dom dom g -> prod_k e. (M...M)gA e. dom dom g)))
41	40	com12 14	. . . . . . 7 |- (M e. ZZ -> (g e. Magma -> (A.k e. (M...M)A e. dom dom g -> prod_k e. (M...M)gA e. dom dom g)))
42		eluz2 7590	. . . . . . . . . . . . 13 |- (m e. (ZZ>=` M) <-> (M e. ZZ /\ m e. ZZ /\ M <_ m))
43		fzssp1 7679	. . . . . . . . . . . . . . 15 |- ((M e. ZZ /\ m e. ZZ) -> (M...m) C_ (M...(m + 1)))
44		ssralv 2672	. . . . . . . . . . . . . . . . . . . 20 |- ((M...m) C_ (M...(m + 1)) -> (A.k e. (M...(m + 1))A e. dom dom g -> A.k e. (M...m)A e. dom dom g))
45		simp3 878	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((A.k e. (M...m)A e. dom dom g /\ ((M...m) C_ (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. dom dom g) /\ g e. Magma) /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) /\ m e. (ZZ>=` M)) -> m e. (ZZ>=` M))
46		simp12r 990	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((A.k e. (M...m)A e. dom dom g /\ ((M...m) C_ (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. dom dom g) /\ g e. Magma) /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) /\ m e. (ZZ>=` M)) -> A.k e. (M...(m + 1))A e. dom dom g)
47		visset 2295	. . . . . . . . . . . . . . . . . . . . . . . 24 |- g e. _V
48	47	fprodp1s 14682	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. dom dom g) -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A))
49	45, 46, 48	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . 22 |- (((A.k e. (M...m)A e. dom dom g /\ ((M...m) C_ (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. dom dom g) /\ g e. Magma) /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) /\ m e. (ZZ>=` M)) -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A))
50	49	3exp 1066	. . . . . . . . . . . . . . . . . . . . 21 |- ((A.k e. (M...m)A e. dom dom g /\ ((M...m) C_ (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. dom dom g) /\ g e. Magma) -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (m e. (ZZ>=` M) -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A))))
51	50	3exp 1066	. . . . . . . . . . . . . . . . . . . 20 |- (A.k e. (M...m)A e. dom dom g -> (((M...m) C_ (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. dom dom g) -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (m e. (ZZ>=` M) -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A))))))
52	44, 51	syl6 25	. . . . . . . . . . . . . . . . . . 19 |- ((M...m) C_ (M...(m + 1)) -> (A.k e. (M...(m + 1))A e. dom dom g -> (((M...m) C_ (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. dom dom g) -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (m e. (ZZ>=` M) -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A)))))))
53	52	imp 377	. . . . . . . . . . . . . . . . . 18 |- (((M...m) C_ (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. dom dom g) -> (((M...m) C_ (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. dom dom g) -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (m e. (ZZ>=` M) -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A))))))
54	53	pm2.43i 78	. . . . . . . . . . . . . . . . 17 |- (((M...m) C_ (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. dom dom g) -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (m e. (ZZ>=` M) -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A)))))
55	54	ex 402	. . . . . . . . . . . . . . . 16 |- ((M...m) C_ (M...(m + 1)) -> (A.k e. (M...(m + 1))A e. dom dom g -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (m e. (ZZ>=` M) -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A))))))
56	55	com25 48	. . . . . . . . . . . . . . 15 |- ((M...m) C_ (M...(m + 1)) -> (m e. (ZZ>=` M) -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A))))))
57	43, 56	syl 12	. . . . . . . . . . . . . 14 |- ((M e. ZZ /\ m e. ZZ) -> (m e. (ZZ>=` M) -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A))))))
58	57	3adant3 896	. . . . . . . . . . . . 13 |- ((M e. ZZ /\ m e. ZZ /\ M <_ m) -> (m e. (ZZ>=` M) -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A))))))
59	42, 58	sylbi 216	. . . . . . . . . . . 12 |- (m e. (ZZ>=` M) -> (m e. (ZZ>=` M) -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A))))))
60	59	pm2.43i 78	. . . . . . . . . . 11 |- (m e. (ZZ>=` M) -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A)))))
61	60	3imp 1061	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ g e. Magma /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) -> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A)))
62		simpl2 880	. . . . . . . . . . . . . 14 |- (((m e. (ZZ>=` M) /\ g e. Magma /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) /\ A.k e. (M...(m + 1))A e. dom dom g) -> g e. Magma)
63		eluzelz 7592	. . . . . . . . . . . . . . . . 17 |- (m e. (ZZ>=` M) -> m e. ZZ)
64		eluzel2 7593	. . . . . . . . . . . . . . . . 17 |- (m e. (ZZ>=` M) -> M e. ZZ)
65	63, 64	jca 310	. . . . . . . . . . . . . . . 16 |- (m e. (ZZ>=` M) -> (m e. ZZ /\ M e. ZZ))
66	43	ancoms 484	. . . . . . . . . . . . . . . . . . 19 |- ((m e. ZZ /\ M e. ZZ) -> (M...m) C_ (M...(m + 1)))
67	66, 44	syl 12	. . . . . . . . . . . . . . . . . 18 |- ((m e. ZZ /\ M e. ZZ) -> (A.k e. (M...(m + 1))A e. dom dom g -> A.k e. (M...m)A e. dom dom g))
68		pm2.27 76	. . . . . . . . . . . . . . . . . . . 20 |- (A.k e. (M...m)A e. dom dom g -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> prod_k e. (M...m)gA e. dom dom g))
69		idd 75	. . . . . . . . . . . . . . . . . . . . . 22 |- (g e. Magma -> (prod_k e. (M...m)gA e. dom dom g -> prod_k e. (M...m)gA e. dom dom g))
70	69	a1i 8	. . . . . . . . . . . . . . . . . . . . 21 |- (m e. (ZZ>=` M) -> (g e. Magma -> (prod_k e. (M...m)gA e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)))
71	70	com13 37	. . . . . . . . . . . . . . . . . . . 20 |- (prod_k e. (M...m)gA e. dom dom g -> (g e. Magma -> (m e. (ZZ>=` M) -> prod_k e. (M...m)gA e. dom dom g)))
72	68, 71	syl6 25	. . . . . . . . . . . . . . . . . . 19 |- (A.k e. (M...m)A e. dom dom g -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (g e. Magma -> (m e. (ZZ>=` M) -> prod_k e. (M...m)gA e. dom dom g))))
73	72	com23 36	. . . . . . . . . . . . . . . . . 18 |- (A.k e. (M...m)A e. dom dom g -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (m e. (ZZ>=` M) -> prod_k e. (M...m)gA e. dom dom g))))
74	67, 73	syl6 25	. . . . . . . . . . . . . . . . 17 |- ((m e. ZZ /\ M e. ZZ) -> (A.k e. (M...(m + 1))A e. dom dom g -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (m e. (ZZ>=` M) -> prod_k e. (M...m)gA e. dom dom g)))))
75	74	com25 48	. . . . . . . . . . . . . . . 16 |- ((m e. ZZ /\ M e. ZZ) -> (m e. (ZZ>=` M) -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)))))
76	65, 75	mpcom 60	. . . . . . . . . . . . . . 15 |- (m e. (ZZ>=` M) -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g))))
77	76	3imp1 1081	. . . . . . . . . . . . . 14 |- (((m e. (ZZ>=` M) /\ g e. Magma /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) /\ A.k e. (M...(m + 1))A e. dom dom g) -> prod_k e. (M...m)gA e. dom dom g)
78		eluzfz2 7659	. . . . . . . . . . . . . . . . 17 |- ((m + 1) e. (ZZ>=` M) -> (m + 1) e. (M...(m + 1)))
79	78	anim1i 361	. . . . . . . . . . . . . . . 16 |- (((m + 1) e. (ZZ>=` M) /\ A.k e. (M...(m + 1))A e. dom dom g) -> ((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. dom dom g))
80		peano2uz 7616	. . . . . . . . . . . . . . . . 17 |- (m e. (ZZ>=` M) -> (m + 1) e. (ZZ>=` M))
81	80	3ad2ant1 897	. . . . . . . . . . . . . . . 16 |- ((m e. (ZZ>=` M) /\ g e. Magma /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) -> (m + 1) e. (ZZ>=` M))
82	79, 81	sylan 497	. . . . . . . . . . . . . . 15 |- (((m e. (ZZ>=` M) /\ g e. Magma /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) /\ A.k e. (M...(m + 1))A e. dom dom g) -> ((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. dom dom g))
83		ra4csbela 2587	. . . . . . . . . . . . . . 15 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))A e. dom dom g) -> [_(m + 1) / k]_A e. dom dom g)
84	82, 83	syl 12	. . . . . . . . . . . . . 14 |- (((m e. (ZZ>=` M) /\ g e. Magma /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) /\ A.k e. (M...(m + 1))A e. dom dom g) -> [_(m + 1) / k]_A e. dom dom g)
85		eqid 1884	. . . . . . . . . . . . . . 15 |- dom dom g = dom dom g
86	85	clmgm 10368	. . . . . . . . . . . . . 14 |- ((g e. Magma /\ prod_k e. (M...m)gA e. dom dom g /\ [_(m + 1) / k]_A e. dom dom g) -> (prod_k e. (M...m)gAg[_(m + 1) / k]_A) e. dom dom g)
87	62, 77, 84, 86	syl111anc 1100	. . . . . . . . . . . . 13 |- (((m e. (ZZ>=` M) /\ g e. Magma /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) /\ A.k e. (M...(m + 1))A e. dom dom g) -> (prod_k e. (M...m)gAg[_(m + 1) / k]_A) e. dom dom g)
88	87	3adant3 896	. . . . . . . . . . . 12 |- (((m e. (ZZ>=` M) /\ g e. Magma /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) /\ A.k e. (M...(m + 1))A e. dom dom g /\ prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A)) -> (prod_k e. (M...m)gAg[_(m + 1) / k]_A) e. dom dom g)
89		eleq1 1957	. . . . . . . . . . . . 13 |- (prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A) -> (prod_k e. (M...(m + 1))gA e. dom dom g <-> (prod_k e. (M...m)gAg[_(m + 1) / k]_A) e. dom dom g))
90	89	3ad2ant3 899	. . . . . . . . . . . 12 |- (((m e. (ZZ>=` M) /\ g e. Magma /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) /\ A.k e. (M...(m + 1))A e. dom dom g /\ prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A)) -> (prod_k e. (M...(m + 1))gA e. dom dom g <-> (prod_k e. (M...m)gAg[_(m + 1) / k]_A) e. dom dom g))
91	88, 90	mpbird 213	. . . . . . . . . . 11 |- (((m e. (ZZ>=` M) /\ g e. Magma /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) /\ A.k e. (M...(m + 1))A e. dom dom g /\ prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A)) -> prod_k e. (M...(m + 1))gA e. dom dom g)
92	91	3exp 1066	. . . . . . . . . 10 |- ((m e. (ZZ>=` M) /\ g e. Magma /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) -> (A.k e. (M...(m + 1))A e. dom dom g -> (prod_k e. (M...(m + 1))gA = (prod_k e. (M...m)gAg[_(m + 1) / k]_A) -> prod_k e. (M...(m + 1))gA e. dom dom g)))
93	61, 92	mpdd 57	. . . . . . . . 9 |- ((m e. (ZZ>=` M) /\ g e. Magma /\ (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) -> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...(m + 1))gA e. dom dom g))
94	93	3exp 1066	. . . . . . . 8 |- (m e. (ZZ>=` M) -> (g e. Magma -> ((A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g) -> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...(m + 1))gA e. dom dom g))))
95	94	a2d 16	. . . . . . 7 |- (m e. (ZZ>=` M) -> ((g e. Magma -> (A.k e. (M...m)A e. dom dom g -> prod_k e. (M...m)gA e. dom dom g)) -> (g e. Magma -> (A.k e. (M...(m + 1))A e. dom dom g -> prod_k e. (M...(m + 1))gA e. dom dom g))))
96	15, 21, 27, 33, 41, 95	uzind4 7619	. . . . . 6 |- (N e. (ZZ>=` M) -> (g e. Magma -> (A.k e. (M...N)A e. dom dom g -> prod_k e. (M...N)gA e. dom dom g)))
97	96	3imp 1061	. . . . 5 |- ((N e. (ZZ>=` M) /\ g e. Magma /\ A.k e. (M...N)A e. dom dom g) -> prod_k e. (M...N)gA e. dom dom g)
98	9, 97	vtoclg 2346	. . . 4 |- (G e. Magma -> ((N e. (ZZ>=` M) /\ G e. Magma /\ A.k e. (M...N)A e. dom dom G) -> prod_k e. (M...N)GA e. dom dom G))
99	98	3ad2ant2 898	. . 3 |- ((N e. (ZZ>=` M) /\ G e. Magma /\ A.k e. (M...N)A e. dom dom G) -> ((N e. (ZZ>=` M) /\ G e. Magma /\ A.k e. (M...N)A e. dom dom G) -> prod_k e. (M...N)GA e. dom dom G))
100	99	pm2.43i 78	. 2 |- ((N e. (ZZ>=` M) /\ G e. Magma /\ A.k e. (M...N)A e. dom dom G) -> prod_k e. (M...N)GA e. dom dom G)
101		clfsebs.1	. . . . . 6 |- X = dom dom G
102	101	eqcomi 1888	. . . . 5 |- dom dom G = X
103	102	eleq2i 1961	. . . 4 |- (A e. dom dom G <-> A e. X)
104	103	ralbii 2127	. . 3 |- (A.k e. (M...N)A e. dom dom G <-> A.k e. (M...N)A e. X)
105	104	3anbi3i 1060	. 2 |- ((N e. (ZZ>=` M) /\ G e. Magma /\ A.k e. (M...N)A e. dom dom G) <-> (N e. (ZZ>=` M) /\ G e. Magma /\ A.k e. (M...N)A e. X))
106	102	eleq2i 1961	. 2 |- (prod_k e. (M...N)GA e. dom dom G <-> prod_k e. (M...N)GA e. X)
107	100, 105, 106	3imtr3i 235	1 |- ((N e. (ZZ>=` M) /\ G e. Magma /\ A.k e. (M...N)A e. X) -> prod_k e. (M...N)GA e. X)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  [_csb 2540   C_ wss 2593   class class class wbr 3338  dom cdm 3986  ` cfv 3998  (class class class)co 4884  1c1 6387   + caddc 6389   <_ cle 6448  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637  Magmacmagm 10365  prod_cprd2 14654

This theorem is referenced by:  clfsebsg 14708  clfsebs2 14710  fprodadd 14713

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-mgm 10366  df-prod 14653  df-prod2 14655

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