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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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