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Theorem taddc 6229
 Description: T raising rule for cardinal sum. (Contributed by SF, 11-Mar-2015.)
Assertion
Ref Expression
taddc (((A NC B NC X NC ) Tc A = ( Tc B +c X)) → c NC X = Tc c)
Distinct variable group:   X,c
Allowed substitution hints:   A(c)   B(c)

Proof of Theorem taddc
Dummy variables a b w x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elncs 6119 . . . . . 6 (A NCx A = Nc x)
2 elncs 6119 . . . . . 6 (B NCy B = Nc y)
3 elncs 6119 . . . . . 6 (X NCz X = Nc z)
41, 2, 33anbi123i 1140 . . . . 5 ((A NC B NC X NC ) ↔ (x A = Nc x y B = Nc y z X = Nc z))
5 eeeanv 1914 . . . . 5 (xyz(A = Nc x B = Nc y X = Nc z) ↔ (x A = Nc x y B = Nc y z X = Nc z))
64, 5bitr4i 243 . . . 4 ((A NC B NC X NC ) ↔ xyz(A = Nc x B = Nc y X = Nc z))
7 vex 2862 . . . . . . . . . . 11 x V
87tcnc 6225 . . . . . . . . . 10 Tc Nc x = Nc 1x
9 vex 2862 . . . . . . . . . . . 12 y V
109tcnc 6225 . . . . . . . . . . 11 Tc Nc y = Nc 1y
1110addceq1i 4386 . . . . . . . . . 10 ( Tc Nc y +c Nc z) = ( Nc 1y +c Nc z)
128, 11eqeq12i 2366 . . . . . . . . 9 ( Tc Nc x = ( Tc Nc y +c Nc z) ↔ Nc 1x = ( Nc 1y +c Nc z))
13 eqcom 2355 . . . . . . . . 9 ( Nc 1x = ( Nc 1y +c Nc z) ↔ ( Nc 1y +c Nc z) = Nc 1x)
149pw1ex 4303 . . . . . . . . . . . 12 1y V
1514ncelncsi 6121 . . . . . . . . . . 11 Nc 1y NC
16 vex 2862 . . . . . . . . . . . 12 z V
1716ncelncsi 6121 . . . . . . . . . . 11 Nc z NC
18 ncaddccl 6144 . . . . . . . . . . 11 (( Nc 1y NC Nc z NC ) → ( Nc 1y +c Nc z) NC )
1915, 17, 18mp2an 653 . . . . . . . . . 10 ( Nc 1y +c Nc z) NC
20 ncseqnc 6128 . . . . . . . . . 10 (( Nc 1y +c Nc z) NC → (( Nc 1y +c Nc z) = Nc 1x1x ( Nc 1y +c Nc z)))
2119, 20ax-mp 5 . . . . . . . . 9 (( Nc 1y +c Nc z) = Nc 1x1x ( Nc 1y +c Nc z))
2212, 13, 213bitri 262 . . . . . . . 8 ( Tc Nc x = ( Tc Nc y +c Nc z) ↔ 1x ( Nc 1y +c Nc z))
23 eladdc 4398 . . . . . . . . 9 (1x ( Nc 1y +c Nc z) ↔ a Nc 1yb Nc z((ab) = 1x = (ab)))
24 vex 2862 . . . . . . . . . . . . . 14 a V
25 vex 2862 . . . . . . . . . . . . . 14 b V
2624, 25pw1equn 4331 . . . . . . . . . . . . 13 (1x = (ab) ↔ cw(x = (cw) a = 1c b = 1w))
27 simp3 957 . . . . . . . . . . . . . . . 16 ((x = (cw) a = 1c b = 1w) → b = 1w)
28 elnc 6125 . . . . . . . . . . . . . . . . 17 (b Nc zbz)
29 ensym 6037 . . . . . . . . . . . . . . . . . 18 (bzzb)
30 breq2 4643 . . . . . . . . . . . . . . . . . . 19 (b = 1w → (zbz1w))
3130biimpcd 215 . . . . . . . . . . . . . . . . . 18 (zb → (b = 1wz1w))
3229, 31sylbi 187 . . . . . . . . . . . . . . . . 17 (bz → (b = 1wz1w))
3328, 32sylbi 187 . . . . . . . . . . . . . . . 16 (b Nc z → (b = 1wz1w))
3427, 33syl5 28 . . . . . . . . . . . . . . 15 (b Nc z → ((x = (cw) a = 1c b = 1w) → z1w))
3534eximdv 1622 . . . . . . . . . . . . . 14 (b Nc z → (w(x = (cw) a = 1c b = 1w) → w z1w))
3635exlimdv 1636 . . . . . . . . . . . . 13 (b Nc z → (cw(x = (cw) a = 1c b = 1w) → w z1w))
3726, 36syl5bi 208 . . . . . . . . . . . 12 (b Nc z → (1x = (ab) → w z1w))
3837adantld 453 . . . . . . . . . . 11 (b Nc z → (((ab) = 1x = (ab)) → w z1w))
3938rexlimiv 2732 . . . . . . . . . 10 (b Nc z((ab) = 1x = (ab)) → w z1w)
4039rexlimivw 2734 . . . . . . . . 9 (a Nc 1yb Nc z((ab) = 1x = (ab)) → w z1w)
4123, 40sylbi 187 . . . . . . . 8 (1x ( Nc 1y +c Nc z) → w z1w)
4222, 41sylbi 187 . . . . . . 7 ( Tc Nc x = ( Tc Nc y +c Nc z) → w z1w)
43 tceq 6158 . . . . . . . . . 10 (A = Nc xTc A = Tc Nc x)
44433ad2ant1 976 . . . . . . . . 9 ((A = Nc x B = Nc y X = Nc z) → Tc A = Tc Nc x)
45 tceq 6158 . . . . . . . . . . . 12 (B = Nc yTc B = Tc Nc y)
4645adantr 451 . . . . . . . . . . 11 ((B = Nc y X = Nc z) → Tc B = Tc Nc y)
47 simpr 447 . . . . . . . . . . 11 ((B = Nc y X = Nc z) → X = Nc z)
4846, 47addceq12d 4391 . . . . . . . . . 10 ((B = Nc y X = Nc z) → ( Tc B +c X) = ( Tc Nc y +c Nc z))
49483adant1 973 . . . . . . . . 9 ((A = Nc x B = Nc y X = Nc z) → ( Tc B +c X) = ( Tc Nc y +c Nc z))
5044, 49eqeq12d 2367 . . . . . . . 8 ((A = Nc x B = Nc y X = Nc z) → ( Tc A = ( Tc B +c X) ↔ Tc Nc x = ( Tc Nc y +c Nc z)))
51 eqeq1 2359 . . . . . . . . . . 11 (X = Nc z → (X = Nc 1wNc z = Nc 1w))
5216eqnc 6127 . . . . . . . . . . 11 ( Nc z = Nc 1wz1w)
5351, 52syl6bb 252 . . . . . . . . . 10 (X = Nc z → (X = Nc 1wz1w))
5453exbidv 1626 . . . . . . . . 9 (X = Nc z → (w X = Nc 1ww z1w))
55543ad2ant3 978 . . . . . . . 8 ((A = Nc x B = Nc y X = Nc z) → (w X = Nc 1ww z1w))
5650, 55imbi12d 311 . . . . . . 7 ((A = Nc x B = Nc y X = Nc z) → (( Tc A = ( Tc B +c X) → w X = Nc 1w) ↔ ( Tc Nc x = ( Tc Nc y +c Nc z) → w z1w)))
5742, 56mpbiri 224 . . . . . 6 ((A = Nc x B = Nc y X = Nc z) → ( Tc A = ( Tc B +c X) → w X = Nc 1w))
5857exlimiv 1634 . . . . 5 (z(A = Nc x B = Nc y X = Nc z) → ( Tc A = ( Tc B +c X) → w X = Nc 1w))
5958exlimivv 1635 . . . 4 (xyz(A = Nc x B = Nc y X = Nc z) → ( Tc A = ( Tc B +c X) → w X = Nc 1w))
606, 59sylbi 187 . . 3 ((A NC B NC X NC ) → ( Tc A = ( Tc B +c X) → w X = Nc 1w))
6160imp 418 . 2 (((A NC B NC X NC ) Tc A = ( Tc B +c X)) → w X = Nc 1w)
62 df-rex 2620 . . 3 (c NC X = Tc cc(c NC X = Tc c))
63 elncs 6119 . . . . . 6 (c NCw c = Nc w)
6463anbi1i 676 . . . . 5 ((c NC X = Tc c) ↔ (w c = Nc w X = Tc c))
65 19.41v 1901 . . . . 5 (w(c = Nc w X = Tc c) ↔ (w c = Nc w X = Tc c))
6664, 65bitr4i 243 . . . 4 ((c NC X = Tc c) ↔ w(c = Nc w X = Tc c))
6766exbii 1582 . . 3 (c(c NC X = Tc c) ↔ cw(c = Nc w X = Tc c))
68 excom 1741 . . . 4 (cw(c = Nc w X = Tc c) ↔ wc(c = Nc w X = Tc c))
69 ncex 6117 . . . . . 6 Nc w V
70 tceq 6158 . . . . . . . 8 (c = Nc wTc c = Tc Nc w)
71 vex 2862 . . . . . . . . 9 w V
7271tcnc 6225 . . . . . . . 8 Tc Nc w = Nc 1w
7370, 72syl6eq 2401 . . . . . . 7 (c = Nc wTc c = Nc 1w)
7473eqeq2d 2364 . . . . . 6 (c = Nc w → (X = Tc cX = Nc 1w))
7569, 74ceqsexv 2894 . . . . 5 (c(c = Nc w X = Tc c) ↔ X = Nc 1w)
7675exbii 1582 . . . 4 (wc(c = Nc w X = Tc c) ↔ w X = Nc 1w)
7768, 76bitri 240 . . 3 (cw(c = Nc w X = Tc c) ↔ w X = Nc 1w)
7862, 67, 773bitri 262 . 2 (c NC X = Tc cw X = Nc 1w)
7961, 78sylibr 203 1 (((A NC B NC X NC ) Tc A = ( Tc B +c X)) → c NC X = Tc c)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  ℘1cpw1 4135   +c cplc 4375   class class class wbr 4639   ≈ cen 6028   NC cncs 6088   Nc cnc 6091   Tc ctc 6093 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-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-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-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  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-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101  df-tc 6103 This theorem is referenced by:  tlecg  6230
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