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 Description: Lemma for addccan2nc 6265. Stratification helper theorem. (Contributed by Scott Fenton, 2-Aug-2019.)
Assertion
Ref Expression
addccan2nclem1 (x( AddC (1st (V × {n})))yy = (x +c n))
Distinct variable group:   x,n,y

Proof of Theorem addccan2nclem1
Dummy variables p q z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brco 4883 . . 3 (x( AddC (1st (V × {n})))yz(x(1st (V × {n}))z z AddC y))
2 brcnv 4892 . . . . . 6 (x(1st (V × {n}))zz(1st (V × {n}))x)
3 brres 4949 . . . . . 6 (z(1st (V × {n}))x ↔ (z1st x z (V × {n})))
4 ancom 437 . . . . . . . . 9 ((z1st x z (V × {n})) ↔ (z (V × {n}) z1st x))
5 elxp2 4802 . . . . . . . . . . 11 (z (V × {n}) ↔ p V q {n}z = p, q)
6 rexv 2873 . . . . . . . . . . 11 (p V q {n}z = p, qpq {n}z = p, q)
7 vex 2862 . . . . . . . . . . . . 13 n V
8 opeq2 4579 . . . . . . . . . . . . . 14 (q = np, q = p, n)
98eqeq2d 2364 . . . . . . . . . . . . 13 (q = n → (z = p, qz = p, n))
107, 9rexsn 3768 . . . . . . . . . . . 12 (q {n}z = p, qz = p, n)
1110exbii 1582 . . . . . . . . . . 11 (pq {n}z = p, qp z = p, n)
125, 6, 113bitri 262 . . . . . . . . . 10 (z (V × {n}) ↔ p z = p, n)
1312anbi1i 676 . . . . . . . . 9 ((z (V × {n}) z1st x) ↔ (p z = p, n z1st x))
144, 13bitri 240 . . . . . . . 8 ((z1st x z (V × {n})) ↔ (p z = p, n z1st x))
15 exancom 1586 . . . . . . . . 9 (p(z1st x z = p, n) ↔ p(z = p, n z1st x))
16 19.41v 1901 . . . . . . . . 9 (p(z = p, n z1st x) ↔ (p z = p, n z1st x))
1715, 16bitri 240 . . . . . . . 8 (p(z1st x z = p, n) ↔ (p z = p, n z1st x))
1814, 17bitr4i 243 . . . . . . 7 ((z1st x z (V × {n})) ↔ p(z1st x z = p, n))
19 vex 2862 . . . . . . . . . . . 12 x V
2019br1st 4858 . . . . . . . . . . 11 (z1st xq z = x, q)
2120anbi1i 676 . . . . . . . . . 10 ((z1st x z = p, n) ↔ (q z = x, q z = p, n))
22 19.41v 1901 . . . . . . . . . 10 (q(z = x, q z = p, n) ↔ (q z = x, q z = p, n))
2321, 22bitr4i 243 . . . . . . . . 9 ((z1st x z = p, n) ↔ q(z = x, q z = p, n))
2423exbii 1582 . . . . . . . 8 (p(z1st x z = p, n) ↔ pq(z = x, q z = p, n))
25 eqeq1 2359 . . . . . . . . . . . . 13 (z = p, n → (z = x, qp, n = x, q))
26 opth 4602 . . . . . . . . . . . . 13 (p, n = x, q ↔ (p = x n = q))
2725, 26syl6bb 252 . . . . . . . . . . . 12 (z = p, n → (z = x, q ↔ (p = x n = q)))
2827pm5.32ri 619 . . . . . . . . . . 11 ((z = x, q z = p, n) ↔ ((p = x n = q) z = p, n))
29 equcom 1680 . . . . . . . . . . . . 13 (n = qq = n)
3029anbi2i 675 . . . . . . . . . . . 12 ((p = x n = q) ↔ (p = x q = n))
3130anbi1i 676 . . . . . . . . . . 11 (((p = x n = q) z = p, n) ↔ ((p = x q = n) z = p, n))
32 opeq2 4579 . . . . . . . . . . . . . . 15 (n = qp, n = p, q)
3332equcoms 1681 . . . . . . . . . . . . . 14 (q = np, n = p, q)
3433adantl 452 . . . . . . . . . . . . 13 ((p = x q = n) → p, n = p, q)
3534eqeq2d 2364 . . . . . . . . . . . 12 ((p = x q = n) → (z = p, nz = p, q))
3635pm5.32i 618 . . . . . . . . . . 11 (((p = x q = n) z = p, n) ↔ ((p = x q = n) z = p, q))
3728, 31, 363bitri 262 . . . . . . . . . 10 ((z = x, q z = p, n) ↔ ((p = x q = n) z = p, q))
38 df-3an 936 . . . . . . . . . 10 ((p = x q = n z = p, q) ↔ ((p = x q = n) z = p, q))
3937, 38bitr4i 243 . . . . . . . . 9 ((z = x, q z = p, n) ↔ (p = x q = n z = p, q))
40392exbii 1583 . . . . . . . 8 (pq(z = x, q z = p, n) ↔ pq(p = x q = n z = p, q))
4124, 40bitri 240 . . . . . . 7 (p(z1st x z = p, n) ↔ pq(p = x q = n z = p, q))
42 opeq1 4578 . . . . . . . . 9 (p = xp, q = x, q)
4342eqeq2d 2364 . . . . . . . 8 (p = x → (z = p, qz = x, q))
44 opeq2 4579 . . . . . . . . 9 (q = nx, q = x, n)
4544eqeq2d 2364 . . . . . . . 8 (q = n → (z = x, qz = x, n))
4619, 7, 43, 45ceqsex2v 2896 . . . . . . 7 (pq(p = x q = n z = p, q) ↔ z = x, n)
4718, 41, 463bitri 262 . . . . . 6 ((z1st x z (V × {n})) ↔ z = x, n)
482, 3, 473bitri 262 . . . . 5 (x(1st (V × {n}))zz = x, n)
4948anbi1i 676 . . . 4 ((x(1st (V × {n}))z z AddC y) ↔ (z = x, n z AddC y))
5049exbii 1582 . . 3 (z(x(1st (V × {n}))z z AddC y) ↔ z(z = x, n z AddC y))
5119, 7opex 4588 . . . 4 x, n V
52 breq1 4642 . . . 4 (z = x, n → (z AddC yx, n AddC y))
5351, 52ceqsexv 2894 . . 3 (z(z = x, n z AddC y) ↔ x, n AddC y)
541, 50, 533bitri 262 . 2 (x( AddC (1st (V × {n})))yx, n AddC y)
5519, 7braddcfn 5826 . 2 (x, n AddC y ↔ (x +c n) = y)
56 eqcom 2355 . 2 ((x +c n) = yy = (x +c n))
5754, 55, 563bitri 262 1 (x( AddC (1st (V × {n})))yy = (x +c n))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859  {csn 3737   +c cplc 4375  ⟨cop 4561   class class class wbr 4639  1st c1st 4717   ∘ ccom 4721   × cxp 4770  ◡ccnv 4771   ↾ cres 4774   AddC caddcfn 5745 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 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-addcfn 5746 This theorem is referenced by:  addccan2nclem2  6264
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