Theorem seqzp2 14716

Description: Value of the arbitrary-based recursive sequence builder at a successor value when the operation G is associative. Compare with seqzp1 7791.

Hypotheses

Ref	Expression
isppm.1	|- X = dom dom G
seqzp2.1	|- G e. SemiGrp
seqzp2.2	|- F:(ZZ>=` M)-->X

Assertion

Ref	Expression
seqzp2	|- (N e. (ZZ>=` M) -> ((<.M, G>. seq F)` (N + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (N + 1))))

Proof of Theorem seqzp2

Step	Hyp	Ref	Expression
1		eluz2 7590	. 2 |- (N e. (ZZ>=` M) <-> (M e. ZZ /\ N e. ZZ /\ M <_ N))
2		opreq1 4889	. . . . 5 |- (j = M -> (j + 1) = (M + 1))
3	2	fveq2d 4685	. . . 4 |- (j = M -> ((<.M, G>. seq F)` (j + 1)) = ((<.M, G>. seq F)` (M + 1)))
4	2	fveq2d 4685	. . . . 5 |- (j = M -> ((<.(M + 1), G>. seq F)` (j + 1)) = ((<.(M + 1), G>. seq F)` (M + 1)))
5	4	opreq2d 4898	. . . 4 |- (j = M -> ((F` M)G((<.(M + 1), G>. seq F)` (j + 1))) = ((F` M)G((<.(M + 1), G>. seq F)` (M + 1))))
6	3, 5	eqeq12d 1899	. . 3 |- (j = M -> (((<.M, G>. seq F)` (j + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (j + 1))) <-> ((<.M, G>. seq F)` (M + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (M + 1)))))
7		opreq1 4889	. . . . 5 |- (j = k -> (j + 1) = (k + 1))
8	7	fveq2d 4685	. . . 4 |- (j = k -> ((<.M, G>. seq F)` (j + 1)) = ((<.M, G>. seq F)` (k + 1)))
9	7	fveq2d 4685	. . . . 5 |- (j = k -> ((<.(M + 1), G>. seq F)` (j + 1)) = ((<.(M + 1), G>. seq F)` (k + 1)))
10	9	opreq2d 4898	. . . 4 |- (j = k -> ((F` M)G((<.(M + 1), G>. seq F)` (j + 1))) = ((F` M)G((<.(M + 1), G>. seq F)` (k + 1))))
11	8, 10	eqeq12d 1899	. . 3 |- (j = k -> (((<.M, G>. seq F)` (j + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (j + 1))) <-> ((<.M, G>. seq F)` (k + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))))
12		opreq1 4889	. . . . 5 |- (j = (k + 1) -> (j + 1) = ((k + 1) + 1))
13	12	fveq2d 4685	. . . 4 |- (j = (k + 1) -> ((<.M, G>. seq F)` (j + 1)) = ((<.M, G>. seq F)` ((k + 1) + 1)))
14	12	fveq2d 4685	. . . . 5 |- (j = (k + 1) -> ((<.(M + 1), G>. seq F)` (j + 1)) = ((<.(M + 1), G>. seq F)` ((k + 1) + 1)))
15	14	opreq2d 4898	. . . 4 |- (j = (k + 1) -> ((F` M)G((<.(M + 1), G>. seq F)` (j + 1))) = ((F` M)G((<.(M + 1), G>. seq F)` ((k + 1) + 1))))
16	13, 15	eqeq12d 1899	. . 3 |- (j = (k + 1) -> (((<.M, G>. seq F)` (j + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (j + 1))) <-> ((<.M, G>. seq F)` ((k + 1) + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` ((k + 1) + 1)))))
17		opreq1 4889	. . . . 5 |- (j = N -> (j + 1) = (N + 1))
18	17	fveq2d 4685	. . . 4 |- (j = N -> ((<.M, G>. seq F)` (j + 1)) = ((<.M, G>. seq F)` (N + 1)))
19	17	fveq2d 4685	. . . . 5 |- (j = N -> ((<.(M + 1), G>. seq F)` (j + 1)) = ((<.(M + 1), G>. seq F)` (N + 1)))
20	19	opreq2d 4898	. . . 4 |- (j = N -> ((F` M)G((<.(M + 1), G>. seq F)` (j + 1))) = ((F` M)G((<.(M + 1), G>. seq F)` (N + 1))))
21	18, 20	eqeq12d 1899	. . 3 |- (j = N -> (((<.M, G>. seq F)` (j + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (j + 1))) <-> ((<.M, G>. seq F)` (N + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (N + 1)))))
22		uzid 7596	. . . . 5 |- (M e. ZZ -> M e. (ZZ>=` M))
23		seqzp2.1	. . . . . . 7 |- G e. SemiGrp
24	23	elisseti 2301	. . . . . 6 |- G e. _V
25		seqzp2.2	. . . . . . 7 |- F:(ZZ>=` M)-->X
26		fvex 4689	. . . . . . 7 |- (ZZ>=` M) e. _V
27		ffn 4562	. . . . . . . 8 |- (F:(ZZ>=` M)-->X -> F Fn (ZZ>=` M))
28		fnex 4535	. . . . . . . . 9 |- ((F Fn (ZZ>=` M) /\ (ZZ>=` M) e. _V) -> F e. _V)
29	28	ex 402	. . . . . . . 8 |- (F Fn (ZZ>=` M) -> ((ZZ>=` M) e. _V -> F e. _V))
30	27, 29	syl 12	. . . . . . 7 |- (F:(ZZ>=` M)-->X -> ((ZZ>=` M) e. _V -> F e. _V))
31	25, 26, 30	mp2 54	. . . . . 6 |- F e. _V
32	24, 31	seqzp1 7791	. . . . 5 |- (M e. (ZZ>=` M) -> ((<.M, G>. seq F)` (M + 1)) = (((<.M, G>. seq F)` M)G(F` (M + 1))))
33	22, 32	syl 12	. . . 4 |- (M e. ZZ -> ((<.M, G>. seq F)` (M + 1)) = (((<.M, G>. seq F)` M)G(F` (M + 1))))
34	24, 31	seqz1 7790	. . . . 5 |- (M e. ZZ -> ((<.M, G>. seq F)` M) = (F` M))
35	34	opreq1d 4897	. . . 4 |- (M e. ZZ -> (((<.M, G>. seq F)` M)G(F` (M + 1))) = ((F` M)G(F` (M + 1))))
36		peano2z 7375	. . . . . . 7 |- (M e. ZZ -> (M + 1) e. ZZ)
37	24, 31	seqz1 7790	. . . . . . 7 |- ((M + 1) e. ZZ -> ((<.(M + 1), G>. seq F)` (M + 1)) = (F` (M + 1)))
38	36, 37	syl 12	. . . . . 6 |- (M e. ZZ -> ((<.(M + 1), G>. seq F)` (M + 1)) = (F` (M + 1)))
39	38	eqcomd 1889	. . . . 5 |- (M e. ZZ -> (F` (M + 1)) = ((<.(M + 1), G>. seq F)` (M + 1)))
40	39	opreq2d 4898	. . . 4 |- (M e. ZZ -> ((F` M)G(F` (M + 1))) = ((F` M)G((<.(M + 1), G>. seq F)` (M + 1))))
41	33, 35, 40	3eqtrd 1929	. . 3 |- (M e. ZZ -> ((<.M, G>. seq F)` (M + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (M + 1))))
42		simp1 876	. . . . . . . . 9 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> M e. ZZ)
43		peano2z 7375	. . . . . . . . . 10 |- (k e. ZZ -> (k + 1) e. ZZ)
44	43	3ad2ant2 898	. . . . . . . . 9 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (k + 1) e. ZZ)
45		zre 7348	. . . . . . . . . . 11 |- (M e. ZZ -> M e. RR)
46	45	3ad2ant1 897	. . . . . . . . . 10 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> M e. RR)
47		zre 7348	. . . . . . . . . . 11 |- (k e. ZZ -> k e. RR)
48	47	3ad2ant2 898	. . . . . . . . . 10 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> k e. RR)
49		simp3 878	. . . . . . . . . 10 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> M <_ k)
50		letrp1 6994	. . . . . . . . . 10 |- ((M e. RR /\ k e. RR /\ M <_ k) -> M <_ (k + 1))
51	46, 48, 49, 50	syl111anc 1100	. . . . . . . . 9 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> M <_ (k + 1))
52	42, 44, 51	3jca 1050	. . . . . . . 8 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (M e. ZZ /\ (k + 1) e. ZZ /\ M <_ (k + 1)))
53		eluz2 7590	. . . . . . . 8 |- ((k + 1) e. (ZZ>=` M) <-> (M e. ZZ /\ (k + 1) e. ZZ /\ M <_ (k + 1)))
54	52, 53	sylibr 217	. . . . . . 7 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (k + 1) e. (ZZ>=` M))
55	24, 31	seqzp1 7791	. . . . . . 7 |- ((k + 1) e. (ZZ>=` M) -> ((<.M, G>. seq F)` ((k + 1) + 1)) = (((<.M, G>. seq F)` (k + 1))G(F` ((k + 1) + 1))))
56	54, 55	syl 12	. . . . . 6 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> ((<.M, G>. seq F)` ((k + 1) + 1)) = (((<.M, G>. seq F)` (k + 1))G(F` ((k + 1) + 1))))
57		opreq1 4889	. . . . . 6 |- (((<.M, G>. seq F)` (k + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (k + 1))) -> (((<.M, G>. seq F)` (k + 1))G(F` ((k + 1) + 1))) = (((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))G(F` ((k + 1) + 1))))
58	56, 57	sylan9eq 1948	. . . . 5 |- (((M e. ZZ /\ k e. ZZ /\ M <_ k) /\ ((<.M, G>. seq F)` (k + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))) -> ((<.M, G>. seq F)` ((k + 1) + 1)) = (((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))G(F` ((k + 1) + 1))))
59		eqid 1884	. . . . . . 7 |- dom dom G = dom dom G
60	59	smgrpass2 14701	. . . . . 6 |- ((G e. SemiGrp /\ ((F` M) e. dom dom G /\ ((<.(M + 1), G>. seq F)` (k + 1)) e. dom dom G /\ (F` ((k + 1) + 1)) e. dom dom G)) -> (((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))G(F` ((k + 1) + 1))) = ((F` M)G(((<.(M + 1), G>. seq F)` (k + 1))G(F` ((k + 1) + 1)))))
61		ffvelrn 4787	. . . . . . . . . . 11 |- ((F:(ZZ>=` M)-->X /\ M e. (ZZ>=` M)) -> (F` M) e. X)
62		isppm.1	. . . . . . . . . . 11 |- X = dom dom G
63	61, 62	syl6eleq 1981	. . . . . . . . . 10 |- ((F:(ZZ>=` M)-->X /\ M e. (ZZ>=` M)) -> (F` M) e. dom dom G)
64	63, 25, 22	sylancr 526	. . . . . . . . 9 |- (M e. ZZ -> (F` M) e. dom dom G)
65	64	3ad2ant1 897	. . . . . . . 8 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (F` M) e. dom dom G)
66	36	3ad2ant1 897	. . . . . . . . . . 11 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (M + 1) e. ZZ)
67	24, 31	seqzres 7803	. . . . . . . . . . . 12 |- ((M + 1) e. ZZ -> (<.(M + 1), G>. seq (F |` (ZZ>=` (M + 1)))) = (<.(M + 1), G>. seq F))
68	67	eqcomd 1889	. . . . . . . . . . 11 |- ((M + 1) e. ZZ -> (<.(M + 1), G>. seq F) = (<.(M + 1), G>. seq (F |` (ZZ>=` (M + 1)))))
69	66, 68	syl 12	. . . . . . . . . 10 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (<.(M + 1), G>. seq F) = (<.(M + 1), G>. seq (F |` (ZZ>=` (M + 1)))))
70	69	fveq1d 4683	. . . . . . . . 9 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> ((<.(M + 1), G>. seq F)` (k + 1)) = ((<.(M + 1), G>. seq (F |` (ZZ>=` (M + 1))))` (k + 1)))
71	45	adantr 425	. . . . . . . . . . . . . 14 |- ((M e. ZZ /\ k e. ZZ) -> M e. RR)
72	47	adantl 424	. . . . . . . . . . . . . 14 |- ((M e. ZZ /\ k e. ZZ) -> k e. RR)
73		1re 6598	. . . . . . . . . . . . . . 15 |- 1 e. RR
74	73	a1i 8	. . . . . . . . . . . . . 14 |- ((M e. ZZ /\ k e. ZZ) -> 1 e. RR)
75		leadd1 6808	. . . . . . . . . . . . . 14 |- ((M e. RR /\ k e. RR /\ 1 e. RR) -> (M <_ k <-> (M + 1) <_ (k + 1)))
76	71, 72, 74, 75	syl111anc 1100	. . . . . . . . . . . . 13 |- ((M e. ZZ /\ k e. ZZ) -> (M <_ k <-> (M + 1) <_ (k + 1)))
77	76	biimp3a 1194	. . . . . . . . . . . 12 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (M + 1) <_ (k + 1))
78	66, 44, 77	3jca 1050	. . . . . . . . . . 11 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> ((M + 1) e. ZZ /\ (k + 1) e. ZZ /\ (M + 1) <_ (k + 1)))
79		eluz2 7590	. . . . . . . . . . 11 |- ((k + 1) e. (ZZ>=` (M + 1)) <-> ((M + 1) e. ZZ /\ (k + 1) e. ZZ /\ (M + 1) <_ (k + 1)))
80	78, 79	sylibr 217	. . . . . . . . . 10 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (k + 1) e. (ZZ>=` (M + 1)))
81		fssres 4582	. . . . . . . . . . 11 |- ((F:(ZZ>=` M)-->dom dom G /\ (ZZ>=` (M + 1)) C_ (ZZ>=` M)) -> (F |` (ZZ>=` (M + 1))):(ZZ>=` (M + 1))-->dom dom G)
82	62	eqcomi 1888	. . . . . . . . . . . . 13 |- dom dom G = X
83		feq3 4553	. . . . . . . . . . . . 13 |- (dom dom G = X -> (F:(ZZ>=` M)-->dom dom G <-> F:(ZZ>=` M)-->X))
84	82, 83	ax-mp 7	. . . . . . . . . . . 12 |- (F:(ZZ>=` M)-->dom dom G <-> F:(ZZ>=` M)-->X)
85	25, 84	mpbir 207	. . . . . . . . . . 11 |- F:(ZZ>=` M)-->dom dom G
86		lep1 6990	. . . . . . . . . . . . . . . 16 |- (M e. RR -> M <_ (M + 1))
87	45, 86	syl 12	. . . . . . . . . . . . . . 15 |- (M e. ZZ -> M <_ (M + 1))
88	87	3ad2ant1 897	. . . . . . . . . . . . . 14 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> M <_ (M + 1))
89	42, 66, 88	3jca 1050	. . . . . . . . . . . . 13 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (M e. ZZ /\ (M + 1) e. ZZ /\ M <_ (M + 1)))
90		eluz2 7590	. . . . . . . . . . . . 13 |- ((M + 1) e. (ZZ>=` M) <-> (M e. ZZ /\ (M + 1) e. ZZ /\ M <_ (M + 1)))
91	89, 90	sylibr 217	. . . . . . . . . . . 12 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (M + 1) e. (ZZ>=` M))
92		uzss 7600	. . . . . . . . . . . 12 |- ((M + 1) e. (ZZ>=` M) -> (ZZ>=` (M + 1)) C_ (ZZ>=` M))
93	91, 92	syl 12	. . . . . . . . . . 11 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (ZZ>=` (M + 1)) C_ (ZZ>=` M))
94	81, 85, 93	sylancr 526	. . . . . . . . . 10 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (F |` (ZZ>=` (M + 1))):(ZZ>=` (M + 1))-->dom dom G)
95	59	smgrpmgm 10382	. . . . . . . . . . . 12 |- (G e. SemiGrp -> G:(dom dom G X. dom dom G)-->dom dom G)
96	23, 95	ax-mp 7	. . . . . . . . . . 11 |- G:(dom dom G X. dom dom G)-->dom dom G
97	96	a1i 8	. . . . . . . . . 10 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> G:(dom dom G X. dom dom G)-->dom dom G)
98		ffun 4565	. . . . . . . . . . . . 13 |- (F:(ZZ>=` M)-->X -> Fun F)
99	25, 98	ax-mp 7	. . . . . . . . . . . 12 |- Fun F
100		fvex 4689	. . . . . . . . . . . 12 |- (ZZ>=` (M + 1)) e. _V
101		resfunexg 4500	. . . . . . . . . . . 12 |- ((Fun F /\ (ZZ>=` (M + 1)) e. _V) -> (F |` (ZZ>=` (M + 1))) e. _V)
102	99, 100, 101	mp2an 761	. . . . . . . . . . 11 |- (F |` (ZZ>=` (M + 1))) e. _V
103	24, 102	seqzcl 7801	. . . . . . . . . 10 |- (((k + 1) e. (ZZ>=` (M + 1)) /\ (F |` (ZZ>=` (M + 1))):(ZZ>=` (M + 1))-->dom dom G /\ G:(dom dom G X. dom dom G)-->dom dom G) -> ((<.(M + 1), G>. seq (F |` (ZZ>=` (M + 1))))` (k + 1)) e. dom dom G)
104	80, 94, 97, 103	syl111anc 1100	. . . . . . . . 9 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> ((<.(M + 1), G>. seq (F |` (ZZ>=` (M + 1))))` (k + 1)) e. dom dom G)
105	70, 104	eqeltrd 1971	. . . . . . . 8 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> ((<.(M + 1), G>. seq F)` (k + 1)) e. dom dom G)
106		ffvelrn 4787	. . . . . . . . 9 |- ((F:(ZZ>=` M)-->dom dom G /\ ((k + 1) + 1) e. (ZZ>=` M)) -> (F` ((k + 1) + 1)) e. dom dom G)
107		feq3 4553	. . . . . . . . . . . 12 |- (X = dom dom G -> (F:(ZZ>=` M)-->X <-> F:(ZZ>=` M)-->dom dom G))
108	107	bicomd 580	. . . . . . . . . . 11 |- (X = dom dom G -> (F:(ZZ>=` M)-->dom dom G <-> F:(ZZ>=` M)-->X))
109	62, 108	ax-mp 7	. . . . . . . . . 10 |- (F:(ZZ>=` M)-->dom dom G <-> F:(ZZ>=` M)-->X)
110	25, 109	mpbir 207	. . . . . . . . 9 |- F:(ZZ>=` M)-->dom dom G
111	43	peano2zdi 7376	. . . . . . . . . . . 12 |- (k e. ZZ -> ((k + 1) + 1) e. ZZ)
112	111	3ad2ant2 898	. . . . . . . . . . 11 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> ((k + 1) + 1) e. ZZ)
113		zre 7348	. . . . . . . . . . . . . 14 |- ((k + 1) e. ZZ -> (k + 1) e. RR)
114	43, 113	syl 12	. . . . . . . . . . . . 13 |- (k e. ZZ -> (k + 1) e. RR)
115	114	3ad2ant2 898	. . . . . . . . . . . 12 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (k + 1) e. RR)
116		letrp1 6994	. . . . . . . . . . . 12 |- ((M e. RR /\ (k + 1) e. RR /\ M <_ (k + 1)) -> M <_ ((k + 1) + 1))
117	46, 115, 51, 116	syl111anc 1100	. . . . . . . . . . 11 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> M <_ ((k + 1) + 1))
118	42, 112, 117	3jca 1050	. . . . . . . . . 10 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (M e. ZZ /\ ((k + 1) + 1) e. ZZ /\ M <_ ((k + 1) + 1)))
119		eluz2 7590	. . . . . . . . . 10 |- (((k + 1) + 1) e. (ZZ>=` M) <-> (M e. ZZ /\ ((k + 1) + 1) e. ZZ /\ M <_ ((k + 1) + 1)))
120	118, 119	sylibr 217	. . . . . . . . 9 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> ((k + 1) + 1) e. (ZZ>=` M))
121	106, 110, 120	sylancr 526	. . . . . . . 8 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (F` ((k + 1) + 1)) e. dom dom G)
122	65, 105, 121	3jca 1050	. . . . . . 7 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> ((F` M) e. dom dom G /\ ((<.(M + 1), G>. seq F)` (k + 1)) e. dom dom G /\ (F` ((k + 1) + 1)) e. dom dom G))
123	122	adantr 425	. . . . . 6 |- (((M e. ZZ /\ k e. ZZ /\ M <_ k) /\ ((<.M, G>. seq F)` (k + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))) -> ((F` M) e. dom dom G /\ ((<.(M + 1), G>. seq F)` (k + 1)) e. dom dom G /\ (F` ((k + 1) + 1)) e. dom dom G))
124	60, 23, 123	sylancr 526	. . . . 5 |- (((M e. ZZ /\ k e. ZZ /\ M <_ k) /\ ((<.M, G>. seq F)` (k + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))) -> (((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))G(F` ((k + 1) + 1))) = ((F` M)G(((<.(M + 1), G>. seq F)` (k + 1))G(F` ((k + 1) + 1)))))
125		eluz2 7590	. . . . . . . . . 10 |- (k e. (ZZ>=` M) <-> (M e. ZZ /\ k e. ZZ /\ M <_ k))
126		eluzp1p1 7604	. . . . . . . . . 10 |- (k e. (ZZ>=` M) -> (k + 1) e. (ZZ>=` (M + 1)))
127	125, 126	sylbir 218	. . . . . . . . 9 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (k + 1) e. (ZZ>=` (M + 1)))
128	127	adantr 425	. . . . . . . 8 |- (((M e. ZZ /\ k e. ZZ /\ M <_ k) /\ ((<.M, G>. seq F)` (k + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))) -> (k + 1) e. (ZZ>=` (M + 1)))
129	24, 31	seqzp1 7791	. . . . . . . 8 |- ((k + 1) e. (ZZ>=` (M + 1)) -> ((<.(M + 1), G>. seq F)` ((k + 1) + 1)) = (((<.(M + 1), G>. seq F)` (k + 1))G(F` ((k + 1) + 1))))
130	128, 129	syl 12	. . . . . . 7 |- (((M e. ZZ /\ k e. ZZ /\ M <_ k) /\ ((<.M, G>. seq F)` (k + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))) -> ((<.(M + 1), G>. seq F)` ((k + 1) + 1)) = (((<.(M + 1), G>. seq F)` (k + 1))G(F` ((k + 1) + 1))))
131	130	eqcomd 1889	. . . . . 6 |- (((M e. ZZ /\ k e. ZZ /\ M <_ k) /\ ((<.M, G>. seq F)` (k + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))) -> (((<.(M + 1), G>. seq F)` (k + 1))G(F` ((k + 1) + 1))) = ((<.(M + 1), G>. seq F)` ((k + 1) + 1)))
132	131	opreq2d 4898	. . . . 5 |- (((M e. ZZ /\ k e. ZZ /\ M <_ k) /\ ((<.M, G>. seq F)` (k + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))) -> ((F` M)G(((<.(M + 1), G>. seq F)` (k + 1))G(F` ((k + 1) + 1)))) = ((F` M)G((<.(M + 1), G>. seq F)` ((k + 1) + 1))))
133	58, 124, 132	3eqtrd 1929	. . . 4 |- (((M e. ZZ /\ k e. ZZ /\ M <_ k) /\ ((<.M, G>. seq F)` (k + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (k + 1)))) -> ((<.M, G>. seq F)` ((k + 1) + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` ((k + 1) + 1))))
134	133	ex 402	. . 3 |- ((M e. ZZ /\ k e. ZZ /\ M <_ k) -> (((<.M, G>. seq F)` (k + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (k + 1))) -> ((<.M, G>. seq F)` ((k + 1) + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` ((k + 1) + 1)))))
135	6, 11, 16, 21, 41, 134	uzind 7417	. 2 |- ((M e. ZZ /\ N e. ZZ /\ M <_ N) -> ((<.M, G>. seq F)` (N + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (N + 1))))
136	1, 135	sylbi 216	1 |- (N e. (ZZ>=` M) -> ((<.M, G>. seq F)` (N + 1)) = ((F` M)G((<.(M + 1), G>. seq F)` (N + 1))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  <.cop 3046   class class class wbr 3338   X. cxp 3984  dom cdm 3986   |` cres 3988  Fun wfun 3992   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  RRcr 6385  1c1 6387   + caddc 6389   <_ cle 6448  ZZcz 6451  ZZ>=cuz 7586   seq cseqz 7774  SemiGrpcsem 10377

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-ass 10360  df-mgm 10366  df-sgr 10378

Copyright terms: Public domain