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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 ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ b = ((n +c n) +c n))

Proof of Theorem nncdiv3lem1
Dummy variables m t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrn2 4897 . 2 (n, b ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ mm, n, b ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))
2 elin 3219 . . . 4 (m, n, b ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ (m, n, b Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) m, n, b (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )))
3 vex 2862 . . . . . . 7 b V
43otelins3 5792 . . . . . 6 (m, n, b Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ↔ m, n ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ))
5 opelcnv 4893 . . . . . 6 (m, n ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ↔ n, m ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ))
6 trtxp 5781 . . . . . . . . . 10 (t(ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd )n, m ↔ (tran (1st ⊗ (1st ∩ 2nd ))n t2nd m))
7 df-br 4640 . . . . . . . . . . . 12 (tran (1st ⊗ (1st ∩ 2nd ))nt, n ran (1st ⊗ (1st ∩ 2nd )))
8 elrn2 4897 . . . . . . . . . . . 12 (t, n ran (1st ⊗ (1st ∩ 2nd )) ↔ mm, t, n (1st ⊗ (1st ∩ 2nd )))
9 vex 2862 . . . . . . . . . . . . . . 15 t V
109proj1ex 4593 . . . . . . . . . . . . . 14 Proj1 t V
1110eqvinc 2966 . . . . . . . . . . . . 13 ( Proj1 t = n, nm(m = Proj1 t m = n, n))
12 opeq 4619 . . . . . . . . . . . . . . 15 t = Proj1 t, Proj2 t
1312breq1i 4646 . . . . . . . . . . . . . 14 (t1st n, n Proj1 t, Proj2 t1st n, n)
149proj2ex 4594 . . . . . . . . . . . . . . 15 Proj2 t V
1510, 14opbr1st 5501 . . . . . . . . . . . . . 14 ( Proj1 t, Proj2 t1st n, n Proj1 t = n, n)
1613, 15bitri 240 . . . . . . . . . . . . 13 (t1st n, n Proj1 t = n, n)
17 oteltxp 5782 . . . . . . . . . . . . . . 15 (m, t, n (1st ⊗ (1st ∩ 2nd )) ↔ (m, t 1st m, n (1st ∩ 2nd )))
18 opelcnv 4893 . . . . . . . . . . . . . . . . 17 (m, t 1stt, m 1st )
19 df-br 4640 . . . . . . . . . . . . . . . . 17 (t1st mt, m 1st )
2012breq1i 4646 . . . . . . . . . . . . . . . . . 18 (t1st m Proj1 t, Proj2 t1st m)
2110, 14opbr1st 5501 . . . . . . . . . . . . . . . . . 18 ( Proj1 t, Proj2 t1st m Proj1 t = m)
22 eqcom 2355 . . . . . . . . . . . . . . . . . 18 ( Proj1 t = mm = Proj1 t)
2320, 21, 223bitri 262 . . . . . . . . . . . . . . . . 17 (t1st mm = Proj1 t)
2418, 19, 233bitr2i 264 . . . . . . . . . . . . . . . 16 (m, t 1stm = Proj1 t)
25 elin 3219 . . . . . . . . . . . . . . . . 17 (m, n (1st ∩ 2nd ) ↔ (m, n 1st m, n 2nd ))
26 df-br 4640 . . . . . . . . . . . . . . . . . 18 (m1st nm, n 1st )
27 df-br 4640 . . . . . . . . . . . . . . . . . 18 (m2nd nm, n 2nd )
2826, 27anbi12i 678 . . . . . . . . . . . . . . . . 17 ((m1st n m2nd n) ↔ (m, n 1st m, n 2nd ))
29 vex 2862 . . . . . . . . . . . . . . . . . 18 n V
3029, 29op1st2nd 5790 . . . . . . . . . . . . . . . . 17 ((m1st n m2nd n) ↔ m = n, n)
3125, 28, 303bitr2i 264 . . . . . . . . . . . . . . . 16 (m, n (1st ∩ 2nd ) ↔ m = n, n)
3224, 31anbi12i 678 . . . . . . . . . . . . . . 15 ((m, t 1st m, n (1st ∩ 2nd )) ↔ (m = Proj1 t m = n, n))
3317, 32bitri 240 . . . . . . . . . . . . . 14 (m, t, n (1st ⊗ (1st ∩ 2nd )) ↔ (m = Proj1 t m = n, n))
3433exbii 1582 . . . . . . . . . . . . 13 (mm, t, n (1st ⊗ (1st ∩ 2nd )) ↔ m(m = Proj1 t m = n, n))
3511, 16, 343bitr4ri 269 . . . . . . . . . . . 12 (mm, t, n (1st ⊗ (1st ∩ 2nd )) ↔ t1st n, n)
367, 8, 353bitri 262 . . . . . . . . . . 11 (tran (1st ⊗ (1st ∩ 2nd ))nt1st n, n)
3736anbi1i 676 . . . . . . . . . 10 ((tran (1st ⊗ (1st ∩ 2nd ))n t2nd m) ↔ (t1st n, n t2nd m))
3829, 29opex 4588 . . . . . . . . . . 11 n, n V
39 vex 2862 . . . . . . . . . . 11 m V
4038, 39op1st2nd 5790 . . . . . . . . . 10 ((t1st n, n t2nd m) ↔ t = n, n, m)
416, 37, 403bitri 262 . . . . . . . . 9 (t(ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd )n, mt = n, n, m)
4241rexbii 2639 . . . . . . . 8 (t AddC t(ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd )n, mt AddC t = n, n, m)
43 elima 4754 . . . . . . . 8 (n, m ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ↔ t AddC t(ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd )n, m)
44 df-br 4640 . . . . . . . . 9 (n, n AddC mn, n, m AddC )
45 risset 2661 . . . . . . . . 9 (n, n, m AddCt AddC t = n, n, m)
4644, 45bitri 240 . . . . . . . 8 (n, n AddC mt AddC t = n, n, m)
4742, 43, 463bitr4i 268 . . . . . . 7 (n, m ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ↔ n, n AddC m)
4829, 29braddcfn 5826 . . . . . . 7 (n, n AddC m ↔ (n +c n) = m)
49 eqcom 2355 . . . . . . 7 ((n +c n) = mm = (n +c n))
5047, 48, 493bitri 262 . . . . . 6 (n, m ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ↔ m = (n +c n))
514, 5, 503bitri 262 . . . . 5 (m, n, b Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ↔ m = (n +c n))
52 elima 4754 . . . . . . 7 (m, n, b (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ) ↔ t AddC t((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd ))m, n, b)
53 trtxp 5781 . . . . . . . . 9 (t((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd ))m, n, b ↔ (t(1st 1st )m t((2nd 1st ) ⊗ 2nd )n, b))
54 trtxp 5781 . . . . . . . . . . 11 (t((2nd 1st ) ⊗ 2nd )n, b ↔ (t(2nd 1st )n t2nd b))
5554anbi2i 675 . . . . . . . . . 10 ((t(1st 1st )m t((2nd 1st ) ⊗ 2nd )n, b) ↔ (t(1st 1st )m (t(2nd 1st )n t2nd b)))
56 anass 630 . . . . . . . . . 10 (((t(1st 1st )m t(2nd 1st )n) t2nd b) ↔ (t(1st 1st )m (t(2nd 1st )n t2nd b)))
5739, 29op1st2nd 5790 . . . . . . . . . . . 12 (( Proj1 t1st m Proj1 t2nd n) ↔ Proj1 t = m, n)
58 brco 4883 . . . . . . . . . . . . . 14 (t(1st 1st )mn(t1st n n1st m))
5912breq1i 4646 . . . . . . . . . . . . . . . . 17 (t1st n Proj1 t, Proj2 t1st n)
6010, 14opbr1st 5501 . . . . . . . . . . . . . . . . 17 ( Proj1 t, Proj2 t1st n Proj1 t = n)
61 eqcom 2355 . . . . . . . . . . . . . . . . 17 ( Proj1 t = nn = Proj1 t)
6259, 60, 613bitri 262 . . . . . . . . . . . . . . . 16 (t1st nn = Proj1 t)
6362anbi1i 676 . . . . . . . . . . . . . . 15 ((t1st n n1st m) ↔ (n = Proj1 t n1st m))
6463exbii 1582 . . . . . . . . . . . . . 14 (n(t1st n n1st m) ↔ n(n = Proj1 t n1st m))
65 breq1 4642 . . . . . . . . . . . . . . 15 (n = Proj1 t → (n1st m Proj1 t1st m))
6610, 65ceqsexv 2894 . . . . . . . . . . . . . 14 (n(n = Proj1 t n1st m) ↔ Proj1 t1st m)
6758, 64, 663bitri 262 . . . . . . . . . . . . 13 (t(1st 1st )m Proj1 t1st m)
68 brco 4883 . . . . . . . . . . . . . 14 (t(2nd 1st )nm(t1st m m2nd n))
6923anbi1i 676 . . . . . . . . . . . . . . 15 ((t1st m m2nd n) ↔ (m = Proj1 t m2nd n))
7069exbii 1582 . . . . . . . . . . . . . 14 (m(t1st m m2nd n) ↔ m(m = Proj1 t m2nd n))
71 breq1 4642 . . . . . . . . . . . . . . 15 (m = Proj1 t → (m2nd n Proj1 t2nd n))
7210, 71ceqsexv 2894 . . . . . . . . . . . . . 14 (m(m = Proj1 t m2nd n) ↔ Proj1 t2nd n)
7368, 70, 723bitri 262 . . . . . . . . . . . . 13 (t(2nd 1st )n Proj1 t2nd n)
7467, 73anbi12i 678 . . . . . . . . . . . 12 ((t(1st 1st )m t(2nd 1st )n) ↔ ( Proj1 t1st m Proj1 t2nd n))
7512breq1i 4646 . . . . . . . . . . . . 13 (t1st m, n Proj1 t, Proj2 t1st m, n)
7610, 14opbr1st 5501 . . . . . . . . . . . . 13 ( Proj1 t, Proj2 t1st m, n Proj1 t = m, n)
7775, 76bitri 240 . . . . . . . . . . . 12 (t1st m, n Proj1 t = m, n)
7857, 74, 773bitr4i 268 . . . . . . . . . . 11 ((t(1st 1st )m t(2nd 1st )n) ↔ t1st m, n)
7978anbi1i 676 . . . . . . . . . 10 (((t(1st 1st )m t(2nd 1st )n) t2nd b) ↔ (t1st m, n t2nd b))
8055, 56, 793bitr2i 264 . . . . . . . . 9 ((t(1st 1st )m t((2nd 1st ) ⊗ 2nd )n, b) ↔ (t1st m, n t2nd b))
8139, 29opex 4588 . . . . . . . . . 10 m, n V
8281, 3op1st2nd 5790 . . . . . . . . 9 ((t1st m, n t2nd b) ↔ t = m, n, b)
8353, 80, 823bitri 262 . . . . . . . 8 (t((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd ))m, n, bt = m, n, b)
8483rexbii 2639 . . . . . . 7 (t AddC t((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd ))m, n, bt AddC t = m, n, b)
8552, 84bitri 240 . . . . . 6 (m, n, b (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ) ↔ t AddC t = m, n, b)
86 df-br 4640 . . . . . . 7 (m, n AddC bm, n, b AddC )
87 risset 2661 . . . . . . 7 (m, n, b AddCt AddC t = m, n, b)
8886, 87bitr2i 241 . . . . . 6 (t AddC t = m, n, bm, n AddC b)
8939, 29braddcfn 5826 . . . . . . 7 (m, n AddC b ↔ (m +c n) = b)
90 eqcom 2355 . . . . . . 7 ((m +c n) = bb = (m +c n))
9189, 90bitri 240 . . . . . 6 (m, n AddC bb = (m +c n))
9285, 88, 913bitri 262 . . . . 5 (m, n, b (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC ) ↔ b = (m +c n))
9351, 92anbi12i 678 . . . 4 ((m, n, b Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) m, n, b (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ (m = (n +c n) b = (m +c n)))
942, 93bitri 240 . . 3 (m, n, b ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ (m = (n +c n) b = (m +c n)))
9594exbii 1582 . 2 (mm, n, b ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ m(m = (n +c n) b = (m +c n)))
9629, 29addcex 4394 . . 3 (n +c n) V
97 addceq1 4383 . . . 4 (m = (n +c n) → (m +c n) = ((n +c n) +c n))
9897eqeq2d 2364 . . 3 (m = (n +c n) → (b = (m +c n) ↔ b = ((n +c n) +c n)))
9996, 98ceqsexv 2894 . 2 (m(m = (n +c n) b = (m +c n)) ↔ b = ((n +c n) +c n))
1001, 95, 993bitri 262 1 (n, b ran ( Ins3 ((ran (1st ⊗ (1st ∩ 2nd )) ⊗ 2nd ) “ AddC ) ∩ (((1st 1st ) ⊗ ((2nd 1st ) ⊗ 2nd )) “ AddC )) ↔ b = ((n +c n) +c n))
Colors of variables: wff setvar class
Syntax hints:  wb 176   wa 358  wex 1541   = wceq 1642   wcel 1710  wrex 2615  cin 3208   +c cplc 4375  cop 4561   Proj1 cproj1 4563   Proj2 cproj2 4564   class class class wbr 4639  1st c1st 4717   ccom 4721  cima 4722  ccnv 4771  ran crn 4773  2nd c2nd 4783  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
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