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Theorem eventfin 4517
 Description: If M is even , then so is Tfin M. Theorem X.1.37 of [Rosser] p. 530. (Contributed by SF, 26-Jan-2015.)
Assertion
Ref Expression
eventfin (M EvenfinTfin M Evenfin )

Proof of Theorem eventfin
Dummy variables m n x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2359 . . . . . 6 (x = M → (x = (n +c n) ↔ M = (n +c n)))
21rexbidv 2635 . . . . 5 (x = M → (n Nn x = (n +c n) ↔ n Nn M = (n +c n)))
3 neeq1 2524 . . . . 5 (x = M → (xM))
42, 3anbi12d 691 . . . 4 (x = M → ((n Nn x = (n +c n) x) ↔ (n Nn M = (n +c n) M)))
5 df-evenfin 4444 . . . 4 Evenfin = {x (n Nn x = (n +c n) x)}
64, 5elab2g 2987 . . 3 (M Evenfin → (M Evenfin ↔ (n Nn M = (n +c n) M)))
76ibi 232 . 2 (M Evenfin → (n Nn M = (n +c n) M))
8 addceq2 4384 . . . . . . . . . . . 12 (n = → (n +c n) = (n +c ))
9 addcnul1 4452 . . . . . . . . . . . 12 (n +c ) =
108, 9syl6eq 2401 . . . . . . . . . . 11 (n = → (n +c n) = )
1110necon3i 2555 . . . . . . . . . 10 ((n +c n) ≠ n)
12 tfinprop 4489 . . . . . . . . . . 11 ((n Nn n) → ( Tfin n Nn x n 1x Tfin n))
1312simpld 445 . . . . . . . . . 10 ((n Nn n) → Tfin n Nn )
1411, 13sylan2 460 . . . . . . . . 9 ((n Nn (n +c n) ≠ ) → Tfin n Nn )
15 tfindi 4496 . . . . . . . . . 10 ((n Nn n Nn (n +c n) ≠ ) → Tfin (n +c n) = ( Tfin n +c Tfin n))
16153anidm12 1239 . . . . . . . . 9 ((n Nn (n +c n) ≠ ) → Tfin (n +c n) = ( Tfin n +c Tfin n))
17 addceq12 4385 . . . . . . . . . . . 12 ((m = Tfin n m = Tfin n) → (m +c m) = ( Tfin n +c Tfin n))
1817anidms 626 . . . . . . . . . . 11 (m = Tfin n → (m +c m) = ( Tfin n +c Tfin n))
1918eqeq2d 2364 . . . . . . . . . 10 (m = Tfin n → ( Tfin (n +c n) = (m +c m) ↔ Tfin (n +c n) = ( Tfin n +c Tfin n)))
2019rspcev 2955 . . . . . . . . 9 (( Tfin n Nn Tfin (n +c n) = ( Tfin n +c Tfin n)) → m Nn Tfin (n +c n) = (m +c m))
2114, 16, 20syl2anc 642 . . . . . . . 8 ((n Nn (n +c n) ≠ ) → m Nn Tfin (n +c n) = (m +c m))
22 nncaddccl 4419 . . . . . . . . . 10 ((n Nn n Nn ) → (n +c n) Nn )
2322anidms 626 . . . . . . . . 9 (n Nn → (n +c n) Nn )
24 tfinnnul 4490 . . . . . . . . 9 (((n +c n) Nn (n +c n) ≠ ) → Tfin (n +c n) ≠ )
2523, 24sylan 457 . . . . . . . 8 ((n Nn (n +c n) ≠ ) → Tfin (n +c n) ≠ )
2621, 25jca 518 . . . . . . 7 ((n Nn (n +c n) ≠ ) → (m Nn Tfin (n +c n) = (m +c m) Tfin (n +c n) ≠ ))
27 tfinex 4485 . . . . . . . 8 Tfin (n +c n) V
28 eqeq1 2359 . . . . . . . . . 10 (x = Tfin (n +c n) → (x = (m +c m) ↔ Tfin (n +c n) = (m +c m)))
2928rexbidv 2635 . . . . . . . . 9 (x = Tfin (n +c n) → (m Nn x = (m +c m) ↔ m Nn Tfin (n +c n) = (m +c m)))
30 neeq1 2524 . . . . . . . . 9 (x = Tfin (n +c n) → (xTfin (n +c n) ≠ ))
3129, 30anbi12d 691 . . . . . . . 8 (x = Tfin (n +c n) → ((m Nn x = (m +c m) x) ↔ (m Nn Tfin (n +c n) = (m +c m) Tfin (n +c n) ≠ )))
32 df-evenfin 4444 . . . . . . . 8 Evenfin = {x (m Nn x = (m +c m) x)}
3327, 31, 32elab2 2988 . . . . . . 7 ( Tfin (n +c n) Evenfin ↔ (m Nn Tfin (n +c n) = (m +c m) Tfin (n +c n) ≠ ))
3426, 33sylibr 203 . . . . . 6 ((n Nn (n +c n) ≠ ) → Tfin (n +c n) Evenfin )
3534ex 423 . . . . 5 (n Nn → ((n +c n) ≠ Tfin (n +c n) Evenfin ))
36 neeq1 2524 . . . . . . . 8 (M = (n +c n) → (M ↔ (n +c n) ≠ ))
37 tfineq 4488 . . . . . . . . 9 (M = (n +c n) → Tfin M = Tfin (n +c n))
3837eleq1d 2419 . . . . . . . 8 (M = (n +c n) → ( Tfin M EvenfinTfin (n +c n) Evenfin ))
3936, 38imbi12d 311 . . . . . . 7 (M = (n +c n) → ((MTfin M Evenfin ) ↔ ((n +c n) ≠ Tfin (n +c n) Evenfin )))
4039biimprd 214 . . . . . 6 (M = (n +c n) → (((n +c n) ≠ Tfin (n +c n) Evenfin ) → (MTfin M Evenfin )))
4140com12 27 . . . . 5 (((n +c n) ≠ Tfin (n +c n) Evenfin ) → (M = (n +c n) → (MTfin M Evenfin )))
4235, 41syl 15 . . . 4 (n Nn → (M = (n +c n) → (MTfin M Evenfin )))
4342rexlimiv 2732 . . 3 (n Nn M = (n +c n) → (MTfin M Evenfin ))
4443imp 418 . 2 ((n Nn M = (n +c n) M) → Tfin M Evenfin )
457, 44syl 15 1 (M EvenfinTfin M Evenfin )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550  ℘1cpw1 4135   Nn cnnc 4373   +c cplc 4375   Tfin ctfin 4435   Evenfin cevenfin 4436 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-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-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-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  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-tfin 4443  df-evenfin 4444 This theorem is referenced by:  vinf  4555
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