Theorem nncdiv3lem1 6276

Description: Lemma for nncdiv3 6278. Set up a helper for stratification. (Contributed by SF, 3-Mar-2015.)

Assertion

Ref	Expression
nncdiv3lem1	|- (<.n, b>. e. ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) <-> b = ((n +o n) +o n))

Proof of Theorem nncdiv3lem1

Dummy variables m t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrn2 4898	. 2 |- (<.n, b>. e. ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) <-> E.m<.m, <.n, b>.>. e. ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))
2		elin 3220	. . . 4 |- (<.m, <.n, b>.>. e. ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) <-> (<.m, <.n, b>.>. e. Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) /\ <.m, <.n, b>.>. e. (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )))
3		vex 2863	. . . . . . 7 |- b e. _V
4	3	otelins3 5793	. . . . . 6 |- (<.m, <.n, b>.>. e. Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) <-> <.m, n>. e. `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ))
5		opelcnv 4894	. . . . . 6 |- (<.m, n>. e. `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) <-> <.n, m>. e. ((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ))
6		trtxp 5782	. . . . . . . . . 10 |- (t(ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)<.n, m>. <-> (tran (`'1st (x) (1st i^i 2nd))n /\ t2ndm))
7		df-br 4641	. . . . . . . . . . . 12 |- (tran (`'1st (x) (1st i^i 2nd))n <-> <.t, n>. e. ran (`'1st (x) (1st i^i 2nd)))
8		elrn2 4898	. . . . . . . . . . . 12 |- (<.t, n>. e. ran (`'1st (x) (1st i^i 2nd)) <-> E.m<.m, <.t, n>.>. e. (`'1st (x) (1st i^i 2nd)))
9		vex 2863	. . . . . . . . . . . . . . 15 |- t e. _V
10	9	proj1ex 4594	. . . . . . . . . . . . . 14 |- Proj1 t e. _V
11	10	eqvinc 2967	. . . . . . . . . . . . 13 |- ( Proj1 t = <.n, n>. <-> E.m(m = Proj1 t /\ m = <.n, n>.))
12		opeq 4620	. . . . . . . . . . . . . . 15 |- t = <. Proj1 t, Proj2 t>.
13	12	breq1i 4647	. . . . . . . . . . . . . 14 |- (t1st<.n, n>. <-> <. Proj1 t, Proj2 t>.1st<.n, n>.)
14	9	proj2ex 4595	. . . . . . . . . . . . . . 15 |- Proj2 t e. _V
15	10, 14	opbr1st 5502	. . . . . . . . . . . . . 14 |- (<. Proj1 t, Proj2 t>.1st<.n, n>. <-> Proj1 t = <.n, n>.)
16	13, 15	bitri 240	. . . . . . . . . . . . 13 |- (t1st<.n, n>. <-> Proj1 t = <.n, n>.)
17		oteltxp 5783	. . . . . . . . . . . . . . 15 |- (<.m, <.t, n>.>. e. (`'1st (x) (1st i^i 2nd)) <-> (<.m, t>. e. `'1st /\ <.m, n>. e. (1st i^i 2nd)))
18		opelcnv 4894	. . . . . . . . . . . . . . . . 17 |- (<.m, t>. e. `'1st <-> <.t, m>. e. 1st)
19		df-br 4641	. . . . . . . . . . . . . . . . 17 |- (t1stm <-> <.t, m>. e. 1st)
20	12	breq1i 4647	. . . . . . . . . . . . . . . . . 18 |- (t1stm <-> <. Proj1 t, Proj2 t>.1stm)
21	10, 14	opbr1st 5502	. . . . . . . . . . . . . . . . . 18 |- (<. Proj1 t, Proj2 t>.1stm <-> Proj1 t = m)
22		eqcom 2355	. . . . . . . . . . . . . . . . . 18 |- ( Proj1 t = m <-> m = Proj1 t)
23	20, 21, 22	3bitri 262	. . . . . . . . . . . . . . . . 17 |- (t1stm <-> m = Proj1 t)
24	18, 19, 23	3bitr2i 264	. . . . . . . . . . . . . . . 16 |- (<.m, t>. e. `'1st <-> m = Proj1 t)
25		elin 3220	. . . . . . . . . . . . . . . . 17 |- (<.m, n>. e. (1st i^i 2nd) <-> (<.m, n>. e. 1st /\ <.m, n>. e. 2nd))
26		df-br 4641	. . . . . . . . . . . . . . . . . 18 |- (m1stn <-> <.m, n>. e. 1st)
27		df-br 4641	. . . . . . . . . . . . . . . . . 18 |- (m2ndn <-> <.m, n>. e. 2nd)
28	26, 27	anbi12i 678	. . . . . . . . . . . . . . . . 17 |- ((m1stn /\ m2ndn) <-> (<.m, n>. e. 1st /\ <.m, n>. e. 2nd))
29		vex 2863	. . . . . . . . . . . . . . . . . 18 |- n e. _V
30	29, 29	op1st2nd 5791	. . . . . . . . . . . . . . . . 17 |- ((m1stn /\ m2ndn) <-> m = <.n, n>.)
31	25, 28, 30	3bitr2i 264	. . . . . . . . . . . . . . . 16 |- (<.m, n>. e. (1st i^i 2nd) <-> m = <.n, n>.)
32	24, 31	anbi12i 678	. . . . . . . . . . . . . . 15 |- ((<.m, t>. e. `'1st /\ <.m, n>. e. (1st i^i 2nd)) <-> (m = Proj1 t /\ m = <.n, n>.))
33	17, 32	bitri 240	. . . . . . . . . . . . . 14 |- (<.m, <.t, n>.>. e. (`'1st (x) (1st i^i 2nd)) <-> (m = Proj1 t /\ m = <.n, n>.))
34	33	exbii 1582	. . . . . . . . . . . . 13 |- (E.m<.m, <.t, n>.>. e. (`'1st (x) (1st i^i 2nd)) <-> E.m(m = Proj1 t /\ m = <.n, n>.))
35	11, 16, 34	3bitr4ri 269	. . . . . . . . . . . 12 |- (E.m<.m, <.t, n>.>. e. (`'1st (x) (1st i^i 2nd)) <-> t1st<.n, n>.)
36	7, 8, 35	3bitri 262	. . . . . . . . . . 11 |- (tran (`'1st (x) (1st i^i 2nd))n <-> t1st<.n, n>.)
37	36	anbi1i 676	. . . . . . . . . 10 |- ((tran (`'1st (x) (1st i^i 2nd))n /\ t2ndm) <-> (t1st<.n, n>. /\ t2ndm))
38	29, 29	opex 4589	. . . . . . . . . . 11 |- <.n, n>. e. _V
39		vex 2863	. . . . . . . . . . 11 |- m e. _V
40	38, 39	op1st2nd 5791	. . . . . . . . . 10 |- ((t1st<.n, n>. /\ t2ndm) <-> t = <.<.n, n>., m>.)
41	6, 37, 40	3bitri 262	. . . . . . . . 9 |- (t(ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)<.n, m>. <-> t = <.<.n, n>., m>.)
42	41	rexbii 2640	. . . . . . . 8 |- (E.t e. AddC t(ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)<.n, m>. <-> E.t e. AddC t = <.<.n, n>., m>.)
43		elima 4755	. . . . . . . 8 |- (<.n, m>. e. ((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) <-> E.t e. AddC t(ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)<.n, m>.)
44		df-br 4641	. . . . . . . . 9 |- (<.n, n>. AddC m <-> <.<.n, n>., m>. e. AddC )
45		risset 2662	. . . . . . . . 9 |- (<.<.n, n>., m>. e. AddC <-> E.t e. AddC t = <.<.n, n>., m>.)
46	44, 45	bitri 240	. . . . . . . 8 |- (<.n, n>. AddC m <-> E.t e. AddC t = <.<.n, n>., m>.)
47	42, 43, 46	3bitr4i 268	. . . . . . 7 |- (<.n, m>. e. ((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) <-> <.n, n>. AddC m)
48	29, 29	braddcfn 5827	. . . . . . 7 |- (<.n, n>. AddC m <-> (n +o n) = m)
49		eqcom 2355	. . . . . . 7 |- ((n +o n) = m <-> m = (n +o n))
50	47, 48, 49	3bitri 262	. . . . . 6 |- (<.n, m>. e. ((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) <-> m = (n +o n))
51	4, 5, 50	3bitri 262	. . . . 5 |- (<.m, <.n, b>.>. e. Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) <-> m = (n +o n))
52		elima 4755	. . . . . . 7 |- (<.m, <.n, b>.>. e. (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ) <-> E.t e. AddC t((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))<.m, <.n, b>.>.)
53		trtxp 5782	. . . . . . . . 9 |- (t((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))<.m, <.n, b>.>. <-> (t(1st o. 1st)m /\ t((2nd o. 1st) (x) 2nd)<.n, b>.))
54		trtxp 5782	. . . . . . . . . . 11 |- (t((2nd o. 1st) (x) 2nd)<.n, b>. <-> (t(2nd o. 1st)n /\ t2ndb))
55	54	anbi2i 675	. . . . . . . . . 10 |- ((t(1st o. 1st)m /\ t((2nd o. 1st) (x) 2nd)<.n, b>.) <-> (t(1st o. 1st)m /\ (t(2nd o. 1st)n /\ t2ndb)))
56		anass 630	. . . . . . . . . 10 |- (((t(1st o. 1st)m /\ t(2nd o. 1st)n) /\ t2ndb) <-> (t(1st o. 1st)m /\ (t(2nd o. 1st)n /\ t2ndb)))
57	39, 29	op1st2nd 5791	. . . . . . . . . . . 12 |- (( Proj1 t1stm /\ Proj1 t2ndn) <-> Proj1 t = <.m, n>.)
58		brco 4884	. . . . . . . . . . . . . 14 |- (t(1st o. 1st)m <-> E.n(t1stn /\ n1stm))
59	12	breq1i 4647	. . . . . . . . . . . . . . . . 17 |- (t1stn <-> <. Proj1 t, Proj2 t>.1stn)
60	10, 14	opbr1st 5502	. . . . . . . . . . . . . . . . 17 |- (<. Proj1 t, Proj2 t>.1stn <-> Proj1 t = n)
61		eqcom 2355	. . . . . . . . . . . . . . . . 17 |- ( Proj1 t = n <-> n = Proj1 t)
62	59, 60, 61	3bitri 262	. . . . . . . . . . . . . . . 16 |- (t1stn <-> n = Proj1 t)
63	62	anbi1i 676	. . . . . . . . . . . . . . 15 |- ((t1stn /\ n1stm) <-> (n = Proj1 t /\ n1stm))
64	63	exbii 1582	. . . . . . . . . . . . . 14 |- (E.n(t1stn /\ n1stm) <-> E.n(n = Proj1 t /\ n1stm))
65		breq1 4643	. . . . . . . . . . . . . . 15 |- (n = Proj1 t -> (n1stm <-> Proj1 t1stm))
66	10, 65	ceqsexv 2895	. . . . . . . . . . . . . 14 |- (E.n(n = Proj1 t /\ n1stm) <-> Proj1 t1stm)
67	58, 64, 66	3bitri 262	. . . . . . . . . . . . 13 |- (t(1st o. 1st)m <-> Proj1 t1stm)
68		brco 4884	. . . . . . . . . . . . . 14 |- (t(2nd o. 1st)n <-> E.m(t1stm /\ m2ndn))
69	23	anbi1i 676	. . . . . . . . . . . . . . 15 |- ((t1stm /\ m2ndn) <-> (m = Proj1 t /\ m2ndn))
70	69	exbii 1582	. . . . . . . . . . . . . 14 |- (E.m(t1stm /\ m2ndn) <-> E.m(m = Proj1 t /\ m2ndn))
71		breq1 4643	. . . . . . . . . . . . . . 15 |- (m = Proj1 t -> (m2ndn <-> Proj1 t2ndn))
72	10, 71	ceqsexv 2895	. . . . . . . . . . . . . 14 |- (E.m(m = Proj1 t /\ m2ndn) <-> Proj1 t2ndn)
73	68, 70, 72	3bitri 262	. . . . . . . . . . . . 13 |- (t(2nd o. 1st)n <-> Proj1 t2ndn)
74	67, 73	anbi12i 678	. . . . . . . . . . . 12 |- ((t(1st o. 1st)m /\ t(2nd o. 1st)n) <-> ( Proj1 t1stm /\ Proj1 t2ndn))
75	12	breq1i 4647	. . . . . . . . . . . . 13 |- (t1st<.m, n>. <-> <. Proj1 t, Proj2 t>.1st<.m, n>.)
76	10, 14	opbr1st 5502	. . . . . . . . . . . . 13 |- (<. Proj1 t, Proj2 t>.1st<.m, n>. <-> Proj1 t = <.m, n>.)
77	75, 76	bitri 240	. . . . . . . . . . . 12 |- (t1st<.m, n>. <-> Proj1 t = <.m, n>.)
78	57, 74, 77	3bitr4i 268	. . . . . . . . . . 11 |- ((t(1st o. 1st)m /\ t(2nd o. 1st)n) <-> t1st<.m, n>.)
79	78	anbi1i 676	. . . . . . . . . 10 |- (((t(1st o. 1st)m /\ t(2nd o. 1st)n) /\ t2ndb) <-> (t1st<.m, n>. /\ t2ndb))
80	55, 56, 79	3bitr2i 264	. . . . . . . . 9 |- ((t(1st o. 1st)m /\ t((2nd o. 1st) (x) 2nd)<.n, b>.) <-> (t1st<.m, n>. /\ t2ndb))
81	39, 29	opex 4589	. . . . . . . . . 10 |- <.m, n>. e. _V
82	81, 3	op1st2nd 5791	. . . . . . . . 9 |- ((t1st<.m, n>. /\ t2ndb) <-> t = <.<.m, n>., b>.)
83	53, 80, 82	3bitri 262	. . . . . . . 8 |- (t((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))<.m, <.n, b>.>. <-> t = <.<.m, n>., b>.)
84	83	rexbii 2640	. . . . . . 7 |- (E.t e. AddC t((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))<.m, <.n, b>.>. <-> E.t e. AddC t = <.<.m, n>., b>.)
85	52, 84	bitri 240	. . . . . 6 |- (<.m, <.n, b>.>. e. (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ) <-> E.t e. AddC t = <.<.m, n>., b>.)
86		df-br 4641	. . . . . . 7 |- (<.m, n>. AddC b <-> <.<.m, n>., b>. e. AddC )
87		risset 2662	. . . . . . 7 |- (<.<.m, n>., b>. e. AddC <-> E.t e. AddC t = <.<.m, n>., b>.)
88	86, 87	bitr2i 241	. . . . . 6 |- (E.t e. AddC t = <.<.m, n>., b>. <-> <.m, n>. AddC b)
89	39, 29	braddcfn 5827	. . . . . . 7 |- (<.m, n>. AddC b <-> (m +o n) = b)
90		eqcom 2355	. . . . . . 7 |- ((m +o n) = b <-> b = (m +o n))
91	89, 90	bitri 240	. . . . . 6 |- (<.m, n>. AddC b <-> b = (m +o n))
92	85, 88, 91	3bitri 262	. . . . 5 |- (<.m, <.n, b>.>. e. (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC ) <-> b = (m +o n))
93	51, 92	anbi12i 678	. . . 4 |- ((<.m, <.n, b>.>. e. Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) /\ <.m, <.n, b>.>. e. (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) <-> (m = (n +o n) /\ b = (m +o n)))
94	2, 93	bitri 240	. . 3 |- (<.m, <.n, b>.>. e. ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) <-> (m = (n +o n) /\ b = (m +o n)))
95	94	exbii 1582	. 2 |- (E.m<.m, <.n, b>.>. e. ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) <-> E.m(m = (n +o n) /\ b = (m +o n)))
96	29, 29	addcex 4395	. . 3 |- (n +o n) e. _V
97		addceq1 4384	. . . 4 |- (m = (n +o n) -> (m +o n) = ((n +o n) +o n))
98	97	eqeq2d 2364	. . 3 |- (m = (n +o n) -> (b = (m +o n) <-> b = ((n +o n) +o n)))
99	96, 98	ceqsexv 2895	. 2 |- (E.m(m = (n +o n) /\ b = (m +o n)) <-> b = ((n +o n) +o n))
100	1, 95, 99	3bitri 262	1 |- (<.n, b>. e. ran ( Ins3 `'((ran (`'1st (x) (1st i^i 2nd)) (x) 2nd)" AddC ) i^i (((1st o. 1st) (x) ((2nd o. 1st) (x) 2nd))" AddC )) <-> b = ((n +o n) +o n))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616   i^i cin 3209   +o cplc 4376  <.cop 4562   Proj1 cproj1 4564   Proj2 cproj2 4565   class class class wbr 4640  1stc1st 4718   o. ccom 4722  "cima 4723  `'ccnv 4772  ran crn 4774  2ndc2nd 4784   (x) ctxp 5736   AddC caddcfn 5746   Ins3 cins3 5752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-addcfn 5747  df-ins3 5753

This theorem is referenced by:  nncdiv3lem2  6277  nnc3n3p1  6279