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Theorem tfindi 4496
 Description: The finite T operation distributes over non-empty cardinal sum. Theorem X.1.32 of [Rosser] p. 529. (Contributed by SF, 26-Jan-2015.)
Assertion
Ref Expression
tfindi ((M Nn N Nn (M +c N) ≠ ) → Tfin (M +c N) = ( Tfin M +c Tfin N))

Proof of Theorem tfindi
Dummy variables a b c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3559 . . 3 ((M +c N) ≠ a a (M +c N))
2 nncaddccl 4419 . . . . . . . 8 ((M Nn N Nn ) → (M +c N) Nn )
3 tfincl 4492 . . . . . . . 8 ((M +c N) NnTfin (M +c N) Nn )
42, 3syl 15 . . . . . . 7 ((M Nn N Nn ) → Tfin (M +c N) Nn )
543adant3 975 . . . . . 6 ((M Nn N Nn a (M +c N)) → Tfin (M +c N) Nn )
6 tfincl 4492 . . . . . . . 8 (M NnTfin M Nn )
7 tfincl 4492 . . . . . . . 8 (N NnTfin N Nn )
8 nncaddccl 4419 . . . . . . . 8 (( Tfin M Nn Tfin N Nn ) → ( Tfin M +c Tfin N) Nn )
96, 7, 8syl2an 463 . . . . . . 7 ((M Nn N Nn ) → ( Tfin M +c Tfin N) Nn )
1093adant3 975 . . . . . 6 ((M Nn N Nn a (M +c N)) → ( Tfin M +c Tfin N) Nn )
1123adant3 975 . . . . . . 7 ((M Nn N Nn a (M +c N)) → (M +c N) Nn )
12 simp3 957 . . . . . . 7 ((M Nn N Nn a (M +c N)) → a (M +c N))
13 tfinpw1 4494 . . . . . . 7 (((M +c N) Nn a (M +c N)) → 1a Tfin (M +c N))
1411, 12, 13syl2anc 642 . . . . . 6 ((M Nn N Nn a (M +c N)) → 1a Tfin (M +c N))
15 eladdc 4398 . . . . . . . 8 (a (M +c N) ↔ b M c N ((bc) = a = (bc)))
16 simplll 734 . . . . . . . . . . . . 13 ((((M Nn N Nn ) (b M c N)) (bc) = ) → M Nn )
17 simplrl 736 . . . . . . . . . . . . 13 ((((M Nn N Nn ) (b M c N)) (bc) = ) → b M)
18 tfinpw1 4494 . . . . . . . . . . . . 13 ((M Nn b M) → 1b Tfin M)
1916, 17, 18syl2anc 642 . . . . . . . . . . . 12 ((((M Nn N Nn ) (b M c N)) (bc) = ) → 1b Tfin M)
20 simpllr 735 . . . . . . . . . . . . 13 ((((M Nn N Nn ) (b M c N)) (bc) = ) → N Nn )
21 simplrr 737 . . . . . . . . . . . . 13 ((((M Nn N Nn ) (b M c N)) (bc) = ) → c N)
22 tfinpw1 4494 . . . . . . . . . . . . 13 ((N Nn c N) → 1c Tfin N)
2320, 21, 22syl2anc 642 . . . . . . . . . . . 12 ((((M Nn N Nn ) (b M c N)) (bc) = ) → 1c Tfin N)
24 pw1eq 4143 . . . . . . . . . . . . . 14 ((bc) = 1(bc) = 1)
25 pw1in 4164 . . . . . . . . . . . . . 14 1(bc) = (1b1c)
26 pw10 4161 . . . . . . . . . . . . . 14 1 =
2724, 25, 263eqtr3g 2408 . . . . . . . . . . . . 13 ((bc) = → (1b1c) = )
2827adantl 452 . . . . . . . . . . . 12 ((((M Nn N Nn ) (b M c N)) (bc) = ) → (1b1c) = )
29 eladdci 4399 . . . . . . . . . . . 12 ((1b Tfin M 1c Tfin N (1b1c) = ) → (1b1c) ( Tfin M +c Tfin N))
3019, 23, 28, 29syl3anc 1182 . . . . . . . . . . 11 ((((M Nn N Nn ) (b M c N)) (bc) = ) → (1b1c) ( Tfin M +c Tfin N))
31 pw1eq 4143 . . . . . . . . . . . . 13 (a = (bc) → 1a = 1(bc))
32 pw1un 4163 . . . . . . . . . . . . 13 1(bc) = (1b1c)
3331, 32syl6eq 2401 . . . . . . . . . . . 12 (a = (bc) → 1a = (1b1c))
3433eleq1d 2419 . . . . . . . . . . 11 (a = (bc) → (1a ( Tfin M +c Tfin N) ↔ (1b1c) ( Tfin M +c Tfin N)))
3530, 34syl5ibrcom 213 . . . . . . . . . 10 ((((M Nn N Nn ) (b M c N)) (bc) = ) → (a = (bc) → 1a ( Tfin M +c Tfin N)))
3635expimpd 586 . . . . . . . . 9 (((M Nn N Nn ) (b M c N)) → (((bc) = a = (bc)) → 1a ( Tfin M +c Tfin N)))
3736rexlimdvva 2745 . . . . . . . 8 ((M Nn N Nn ) → (b M c N ((bc) = a = (bc)) → 1a ( Tfin M +c Tfin N)))
3815, 37syl5bi 208 . . . . . . 7 ((M Nn N Nn ) → (a (M +c N) → 1a ( Tfin M +c Tfin N)))
39383impia 1148 . . . . . 6 ((M Nn N Nn a (M +c N)) → 1a ( Tfin M +c Tfin N))
40 nnceleq 4430 . . . . . 6 ((( Tfin (M +c N) Nn ( Tfin M +c Tfin N) Nn ) (1a Tfin (M +c N) 1a ( Tfin M +c Tfin N))) → Tfin (M +c N) = ( Tfin M +c Tfin N))
415, 10, 14, 39, 40syl22anc 1183 . . . . 5 ((M Nn N Nn a (M +c N)) → Tfin (M +c N) = ( Tfin M +c Tfin N))
42413expia 1153 . . . 4 ((M Nn N Nn ) → (a (M +c N) → Tfin (M +c N) = ( Tfin M +c Tfin N)))
4342exlimdv 1636 . . 3 ((M Nn N Nn ) → (a a (M +c N) → Tfin (M +c N) = ( Tfin M +c Tfin N)))
441, 43syl5bi 208 . 2 ((M Nn N Nn ) → ((M +c N) ≠ Tfin (M +c N) = ( Tfin M +c Tfin N)))
45443impia 1148 1 ((M Nn N Nn (M +c N) ≠ ) → Tfin (M +c N) = ( Tfin M +c Tfin N))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  ℘1cpw1 4135   Nn cnnc 4373   +c cplc 4375   Tfin ctfin 4435 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 This theorem is referenced by:  tfinltfinlem1  4500  eventfin  4517  oddtfin  4518  sfintfin  4532
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