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Theorem addccan2nclem2 6264
 Description: Lemma for addccan2nc 6265. Establish stratification for induction. (Contributed by Scott Fenton, 2-Aug-2019.)
Assertion
Ref Expression
addccan2nclem2 ((N V P W) → {x ((x +c N) = (x +c P) → N = P)} V)
Distinct variable groups:   x,N   x,P
Allowed substitution hints:   V(x)   W(x)

Proof of Theorem addccan2nclem2
Dummy variables n p y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unab 3521 . . 3 ({x ¬ (x +c N) = (x +c P)} ∪ {x N = P}) = {x (¬ (x +c N) = (x +c P) N = P)}
2 complab 3524 . . . 4 ∼ {x (x +c N) = (x +c P)} = {x ¬ (x +c N) = (x +c P)}
32uneq1i 3414 . . 3 ( ∼ {x (x +c N) = (x +c P)} ∪ {x N = P}) = ({x ¬ (x +c N) = (x +c P)} ∪ {x N = P})
4 imor 401 . . . 4 (((x +c N) = (x +c P) → N = P) ↔ (¬ (x +c N) = (x +c P) N = P))
54abbii 2465 . . 3 {x ((x +c N) = (x +c P) → N = P)} = {x (¬ (x +c N) = (x +c P) N = P)}
61, 3, 53eqtr4i 2383 . 2 ( ∼ {x (x +c N) = (x +c P)} ∪ {x N = P}) = {x ((x +c N) = (x +c P) → N = P)}
7 addceq2 4384 . . . . . . . 8 (n = N → (x +c n) = (x +c N))
87eqeq1d 2361 . . . . . . 7 (n = N → ((x +c n) = (x +c p) ↔ (x +c N) = (x +c p)))
98abbidv 2467 . . . . . 6 (n = N → {x (x +c n) = (x +c p)} = {x (x +c N) = (x +c p)})
109eleq1d 2419 . . . . 5 (n = N → ({x (x +c n) = (x +c p)} V ↔ {x (x +c N) = (x +c p)} V))
11 addceq2 4384 . . . . . . . 8 (p = P → (x +c p) = (x +c P))
1211eqeq2d 2364 . . . . . . 7 (p = P → ((x +c N) = (x +c p) ↔ (x +c N) = (x +c P)))
1312abbidv 2467 . . . . . 6 (p = P → {x (x +c N) = (x +c p)} = {x (x +c N) = (x +c P)})
1413eleq1d 2419 . . . . 5 (p = P → ({x (x +c N) = (x +c p)} V ↔ {x (x +c N) = (x +c P)} V))
15 elfix 5787 . . . . . . . 8 (x Fix (( AddC (1st (V × {p}))) ( AddC (1st (V × {n})))) ↔ x(( AddC (1st (V × {p}))) ( AddC (1st (V × {n}))))x)
16 brco 4883 . . . . . . . . 9 (x(( AddC (1st (V × {p}))) ( AddC (1st (V × {n}))))xy(x( AddC (1st (V × {n})))y y( AddC (1st (V × {p})))x))
17 addccan2nclem1 6263 . . . . . . . . . . 11 (x( AddC (1st (V × {n})))yy = (x +c n))
18 brcnv 4892 . . . . . . . . . . . 12 (y( AddC (1st (V × {p})))xx( AddC (1st (V × {p})))y)
19 addccan2nclem1 6263 . . . . . . . . . . . 12 (x( AddC (1st (V × {p})))yy = (x +c p))
2018, 19bitri 240 . . . . . . . . . . 11 (y( AddC (1st (V × {p})))xy = (x +c p))
2117, 20anbi12i 678 . . . . . . . . . 10 ((x( AddC (1st (V × {n})))y y( AddC (1st (V × {p})))x) ↔ (y = (x +c n) y = (x +c p)))
2221exbii 1582 . . . . . . . . 9 (y(x( AddC (1st (V × {n})))y y( AddC (1st (V × {p})))x) ↔ y(y = (x +c n) y = (x +c p)))
2316, 22bitri 240 . . . . . . . 8 (x(( AddC (1st (V × {p}))) ( AddC (1st (V × {n}))))xy(y = (x +c n) y = (x +c p)))
24 vex 2862 . . . . . . . . . 10 x V
25 vex 2862 . . . . . . . . . 10 n V
2624, 25addcex 4394 . . . . . . . . 9 (x +c n) V
27 eqeq1 2359 . . . . . . . . 9 (y = (x +c n) → (y = (x +c p) ↔ (x +c n) = (x +c p)))
2826, 27ceqsexv 2894 . . . . . . . 8 (y(y = (x +c n) y = (x +c p)) ↔ (x +c n) = (x +c p))
2915, 23, 283bitri 262 . . . . . . 7 (x Fix (( AddC (1st (V × {p}))) ( AddC (1st (V × {n})))) ↔ (x +c n) = (x +c p))
3029abbi2i 2464 . . . . . 6 Fix (( AddC (1st (V × {p}))) ( AddC (1st (V × {n})))) = {x (x +c n) = (x +c p)}
31 addcfnex 5824 . . . . . . . . . 10 AddC V
32 1stex 4739 . . . . . . . . . . . 12 1st V
33 vvex 4109 . . . . . . . . . . . . 13 V V
34 snex 4111 . . . . . . . . . . . . 13 {p} V
3533, 34xpex 5115 . . . . . . . . . . . 12 (V × {p}) V
3632, 35resex 5117 . . . . . . . . . . 11 (1st (V × {p})) V
3736cnvex 5102 . . . . . . . . . 10 (1st (V × {p})) V
3831, 37coex 4750 . . . . . . . . 9 ( AddC (1st (V × {p}))) V
3938cnvex 5102 . . . . . . . 8 ( AddC (1st (V × {p}))) V
40 snex 4111 . . . . . . . . . . . 12 {n} V
4133, 40xpex 5115 . . . . . . . . . . 11 (V × {n}) V
4232, 41resex 5117 . . . . . . . . . 10 (1st (V × {n})) V
4342cnvex 5102 . . . . . . . . 9 (1st (V × {n})) V
4431, 43coex 4750 . . . . . . . 8 ( AddC (1st (V × {n}))) V
4539, 44coex 4750 . . . . . . 7 (( AddC (1st (V × {p}))) ( AddC (1st (V × {n})))) V
4645fixex 5789 . . . . . 6 Fix (( AddC (1st (V × {p}))) ( AddC (1st (V × {n})))) V
4730, 46eqeltrri 2424 . . . . 5 {x (x +c n) = (x +c p)} V
4810, 14, 47vtocl2g 2918 . . . 4 ((N V P W) → {x (x +c N) = (x +c P)} V)
49 complexg 4099 . . . 4 ({x (x +c N) = (x +c P)} V → ∼ {x (x +c N) = (x +c P)} V)
5048, 49syl 15 . . 3 ((N V P W) → ∼ {x (x +c N) = (x +c P)} V)
51 abexv 4324 . . 3 {x N = P} V
52 unexg 4101 . . 3 (( ∼ {x (x +c N) = (x +c P)} V {x N = P} V) → ( ∼ {x (x +c N) = (x +c P)} ∪ {x N = P}) V)
5350, 51, 52sylancl 643 . 2 ((N V P W) → ( ∼ {x (x +c N) = (x +c P)} ∪ {x N = P}) V)
546, 53syl5eqelr 2438 1 ((N V P W) → {x ((x +c N) = (x +c P) → N = P)} V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859   ∼ ccompl 3205   ∪ cun 3207  {csn 3737   +c cplc 4375   class class class wbr 4639  1st c1st 4717   ∘ ccom 4721   × cxp 4770  ◡ccnv 4771   ↾ cres 4774   Fix cfix 5739   AddC caddcfn 5745 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-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-fo 4793  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-fix 5740  df-cup 5742  df-disj 5744  df-addcfn 5746  df-ins2 5750  df-ins3 5752  df-ins4 5756  df-si3 5758 This theorem is referenced by:  addccan2nc  6265
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