Theorem nncdiv3lem1 6275

Description: Lemma for nncdiv3 6277. 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 4897	. 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 3219	. . . 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 2862	. . . . . . 7 |- b e. _V
4	3	otelins3 5792	. . . . . 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 4893	. . . . . 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 5781	. . . . . . . . . 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 4640	. . . . . . . . . . . 12 |- (tran (`'1st (x) (1st i^i 2nd))n <-> <.t, n>. e. ran (`'1st (x) (1st i^i 2nd)))
8		elrn2 4897	. . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . 15 |- t e. _V
10	9	proj1ex 4593	. . . . . . . . . . . . . 14 |- Proj1 t e. _V
11	10	eqvinc 2966	. . . . . . . . . . . . 13 |- ( Proj1 t = <.n, n>. <-> E.m(m = Proj1 t /\ m = <.n, n>.))
12		opeq 4619	. . . . . . . . . . . . . . 15 |- t = <. Proj1 t, Proj2 t>.
13	12	breq1i 4646	. . . . . . . . . . . . . 14 |- (t1st<.n, n>. <-> <. Proj1 t, Proj2 t>.1st<.n, n>.)
14	9	proj2ex 4594	. . . . . . . . . . . . . . 15 |- Proj2 t e. _V
15	10, 14	opbr1st 5501	. . . . . . . . . . . . . 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 5782	. . . . . . . . . . . . . . 15 |- (<.m, <.t, n>.>. e. (`'1st (x) (1st i^i 2nd)) <-> (<.m, t>. e. `'1st /\ <.m, n>. e. (1st i^i 2nd)))
18		opelcnv 4893	. . . . . . . . . . . . . . . . 17 |- (<.m, t>. e. `'1st <-> <.t, m>. e. 1st)
19		df-br 4640	. . . . . . . . . . . . . . . . 17 |- (t1stm <-> <.t, m>. e. 1st)
20	12	breq1i 4646	. . . . . . . . . . . . . . . . . 18 |- (t1stm <-> <. Proj1 t, Proj2 t>.1stm)
21	10, 14	opbr1st 5501	. . . . . . . . . . . . . . . . . 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 3219	. . . . . . . . . . . . . . . . 17 |- (<.m, n>. e. (1st i^i 2nd) <-> (<.m, n>. e. 1st /\ <.m, n>. e. 2nd))
26		df-br 4640	. . . . . . . . . . . . . . . . . 18 |- (m1stn <-> <.m, n>. e. 1st)
27		df-br 4640	. . . . . . . . . . . . . . . . . 18 |- (m2ndn <-> <.m, n>. e. 2nd)
28	26, 27	anbi12i 678	. . . . . . . . . . . . . . . . 17 |- ((m1stn /\ m2ndn) <-> (<.m, n>. e. 1st /\ <.m, n>. e. 2nd))
29		vex 2862	. . . . . . . . . . . . . . . . . 18 |- n e. _V
30	29, 29	op1st2nd 5790	. . . . . . . . . . . . . . . . 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 4588	. . . . . . . . . . 11 |- <.n, n>. e. _V
39		vex 2862	. . . . . . . . . . 11 |- m e. _V
40	38, 39	op1st2nd 5790	. . . . . . . . . 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 2639	. . . . . . . 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 4754	. . . . . . . 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 4640	. . . . . . . . 9 |- (<.n, n>. AddC m <-> <.<.n, n>., m>. e. AddC )
45		risset 2661	. . . . . . . . 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 5826	. . . . . . 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 4754	. . . . . . 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 5781	. . . . . . . . 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 5781	. . . . . . . . . . 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 5790	. . . . . . . . . . . 12 |- (( Proj1 t1stm /\ Proj1 t2ndn) <-> Proj1 t = <.m, n>.)
58		brco 4883	. . . . . . . . . . . . . 14 |- (t(1st o. 1st)m <-> E.n(t1stn /\ n1stm))
59	12	breq1i 4646	. . . . . . . . . . . . . . . . 17 |- (t1stn <-> <. Proj1 t, Proj2 t>.1stn)
60	10, 14	opbr1st 5501	. . . . . . . . . . . . . . . . 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 4642	. . . . . . . . . . . . . . 15 |- (n = Proj1 t -> (n1stm <-> Proj1 t1stm))
66	10, 65	ceqsexv 2894	. . . . . . . . . . . . . 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 4883	. . . . . . . . . . . . . 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 4642	. . . . . . . . . . . . . . 15 |- (m = Proj1 t -> (m2ndn <-> Proj1 t2ndn))
72	10, 71	ceqsexv 2894	. . . . . . . . . . . . . 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 4646	. . . . . . . . . . . . 13 |- (t1st<.m, n>. <-> <. Proj1 t, Proj2 t>.1st<.m, n>.)
76	10, 14	opbr1st 5501	. . . . . . . . . . . . 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 4588	. . . . . . . . . 10 |- <.m, n>. e. _V
82	81, 3	op1st2nd 5790	. . . . . . . . 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 2639	. . . . . . 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 4640	. . . . . . 7 |- (<.m, n>. AddC b <-> <.<.m, n>., b>. e. AddC )
87		risset 2661	. . . . . . 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 5826	. . . . . . 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 4394	. . 3 |- (n +o n) e. _V
97		addceq1 4383	. . . 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 2894	. 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 2615   i^i cin 3208   +o cplc 4375  <.cop 4561   Proj1 cproj1 4563   Proj2 cproj2 4564   class class class wbr 4639  1stc1st 4717   o. ccom 4721  "cima 4722  `'ccnv 4771  ran crn 4773  2ndc2nd 4783   (x) ctxp 5735   AddC caddcfn 5745   Ins3 cins3 5751

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-addcfn 5746  df-ins3 5752

This theorem is referenced by:  nncdiv3lem2  6276  nnc3n3p1  6278