Theorem addccan2nc 6266

Description: Cancellation law for addition over the cardinal numbers. Biconditional form of theorem XI.3.2 of [Rosser] p. 391. (Contributed by Scott Fenton, 2-Aug-2019.)

Assertion

Ref	Expression
addccan2nc	⊢ ((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) → ((M +c N) = (M +c P) ↔ N = P))

Proof of Theorem addccan2nc

Dummy variables x m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addccan2nclem2 6265	. . . 4 ⊢ ((N ∈ NC ∧ P ∈ NC ) → {x ∣ ((x +c N) = (x +c P) → N = P)} ∈ V)
2		addceq1 4384	. . . . . 6 ⊢ (x = 0c → (x +c N) = (0c +c N))
3		addceq1 4384	. . . . . 6 ⊢ (x = 0c → (x +c P) = (0c +c P))
4	2, 3	eqeq12d 2367	. . . . 5 ⊢ (x = 0c → ((x +c N) = (x +c P) ↔ (0c +c N) = (0c +c P)))
5	4	imbi1d 308	. . . 4 ⊢ (x = 0c → (((x +c N) = (x +c P) → N = P) ↔ ((0c +c N) = (0c +c P) → N = P)))
6		addceq1 4384	. . . . . 6 ⊢ (x = m → (x +c N) = (m +c N))
7		addceq1 4384	. . . . . 6 ⊢ (x = m → (x +c P) = (m +c P))
8	6, 7	eqeq12d 2367	. . . . 5 ⊢ (x = m → ((x +c N) = (x +c P) ↔ (m +c N) = (m +c P)))
9	8	imbi1d 308	. . . 4 ⊢ (x = m → (((x +c N) = (x +c P) → N = P) ↔ ((m +c N) = (m +c P) → N = P)))
10		addceq1 4384	. . . . . 6 ⊢ (x = (m +c 1c) → (x +c N) = ((m +c 1c) +c N))
11		addceq1 4384	. . . . . 6 ⊢ (x = (m +c 1c) → (x +c P) = ((m +c 1c) +c P))
12	10, 11	eqeq12d 2367	. . . . 5 ⊢ (x = (m +c 1c) → ((x +c N) = (x +c P) ↔ ((m +c 1c) +c N) = ((m +c 1c) +c P)))
13	12	imbi1d 308	. . . 4 ⊢ (x = (m +c 1c) → (((x +c N) = (x +c P) → N = P) ↔ (((m +c 1c) +c N) = ((m +c 1c) +c P) → N = P)))
14		addceq1 4384	. . . . . 6 ⊢ (x = M → (x +c N) = (M +c N))
15		addceq1 4384	. . . . . 6 ⊢ (x = M → (x +c P) = (M +c P))
16	14, 15	eqeq12d 2367	. . . . 5 ⊢ (x = M → ((x +c N) = (x +c P) ↔ (M +c N) = (M +c P)))
17	16	imbi1d 308	. . . 4 ⊢ (x = M → (((x +c N) = (x +c P) → N = P) ↔ ((M +c N) = (M +c P) → N = P)))
18		addcid2 4408	. . . . . . 7 ⊢ (0c +c N) = N
19		addcid2 4408	. . . . . . 7 ⊢ (0c +c P) = P
20	18, 19	eqeq12i 2366	. . . . . 6 ⊢ ((0c +c N) = (0c +c P) ↔ N = P)
21	20	biimpi 186	. . . . 5 ⊢ ((0c +c N) = (0c +c P) → N = P)
22	21	a1i 10	. . . 4 ⊢ ((N ∈ NC ∧ P ∈ NC ) → ((0c +c N) = (0c +c P) → N = P))
23		addc32 4417	. . . . . . 7 ⊢ ((m +c 1c) +c N) = ((m +c N) +c 1c)
24		addc32 4417	. . . . . . 7 ⊢ ((m +c 1c) +c P) = ((m +c P) +c 1c)
25	23, 24	eqeq12i 2366	. . . . . 6 ⊢ (((m +c 1c) +c N) = ((m +c 1c) +c P) ↔ ((m +c N) +c 1c) = ((m +c P) +c 1c))
26		nnnc 6147	. . . . . . . . . . 11 ⊢ (m ∈ Nn → m ∈ NC )
27		ncaddccl 6145	. . . . . . . . . . 11 ⊢ ((m ∈ NC ∧ N ∈ NC ) → (m +c N) ∈ NC )
28	26, 27	sylan 457	. . . . . . . . . 10 ⊢ ((m ∈ Nn ∧ N ∈ NC ) → (m +c N) ∈ NC )
29	28	adantrr 697	. . . . . . . . 9 ⊢ ((m ∈ Nn ∧ (N ∈ NC ∧ P ∈ NC )) → (m +c N) ∈ NC )
30	29	adantr 451	. . . . . . . 8 ⊢ (((m ∈ Nn ∧ (N ∈ NC ∧ P ∈ NC )) ∧ ((m +c N) = (m +c P) → N = P)) → (m +c N) ∈ NC )
31		ncaddccl 6145	. . . . . . . . . . 11 ⊢ ((m ∈ NC ∧ P ∈ NC ) → (m +c P) ∈ NC )
32	26, 31	sylan 457	. . . . . . . . . 10 ⊢ ((m ∈ Nn ∧ P ∈ NC ) → (m +c P) ∈ NC )
33	32	adantrl 696	. . . . . . . . 9 ⊢ ((m ∈ Nn ∧ (N ∈ NC ∧ P ∈ NC )) → (m +c P) ∈ NC )
34	33	adantr 451	. . . . . . . 8 ⊢ (((m ∈ Nn ∧ (N ∈ NC ∧ P ∈ NC )) ∧ ((m +c N) = (m +c P) → N = P)) → (m +c P) ∈ NC )
35		peano4nc 6151	. . . . . . . . 9 ⊢ (((m +c N) ∈ NC ∧ (m +c P) ∈ NC ) → (((m +c N) +c 1c) = ((m +c P) +c 1c) ↔ (m +c N) = (m +c P)))
36	35	biimpd 198	. . . . . . . 8 ⊢ (((m +c N) ∈ NC ∧ (m +c P) ∈ NC ) → (((m +c N) +c 1c) = ((m +c P) +c 1c) → (m +c N) = (m +c P)))
37	30, 34, 36	syl2anc 642	. . . . . . 7 ⊢ (((m ∈ Nn ∧ (N ∈ NC ∧ P ∈ NC )) ∧ ((m +c N) = (m +c P) → N = P)) → (((m +c N) +c 1c) = ((m +c P) +c 1c) → (m +c N) = (m +c P)))
38		simpr 447	. . . . . . 7 ⊢ (((m ∈ Nn ∧ (N ∈ NC ∧ P ∈ NC )) ∧ ((m +c N) = (m +c P) → N = P)) → ((m +c N) = (m +c P) → N = P))
39	37, 38	syld 40	. . . . . 6 ⊢ (((m ∈ Nn ∧ (N ∈ NC ∧ P ∈ NC )) ∧ ((m +c N) = (m +c P) → N = P)) → (((m +c N) +c 1c) = ((m +c P) +c 1c) → N = P))
40	25, 39	syl5bi 208	. . . . 5 ⊢ (((m ∈ Nn ∧ (N ∈ NC ∧ P ∈ NC )) ∧ ((m +c N) = (m +c P) → N = P)) → (((m +c 1c) +c N) = ((m +c 1c) +c P) → N = P))
41	40	ex 423	. . . 4 ⊢ ((m ∈ Nn ∧ (N ∈ NC ∧ P ∈ NC )) → (((m +c N) = (m +c P) → N = P) → (((m +c 1c) +c N) = ((m +c 1c) +c P) → N = P)))
42	1, 5, 9, 13, 17, 22, 41	findsd 4411	. . 3 ⊢ ((M ∈ Nn ∧ (N ∈ NC ∧ P ∈ NC )) → ((M +c N) = (M +c P) → N = P))
43	42	3impb 1147	. 2 ⊢ ((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) → ((M +c N) = (M +c P) → N = P))
44		addceq2 4385	. 2 ⊢ (N = P → (M +c N) = (M +c P))
45	43, 44	impbid1 194	1 ⊢ ((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) → ((M +c N) = (M +c P) ↔ N = P))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Vcvv 2860  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   NC cncs 6089

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-fix 5741  df-cup 5743  df-disj 5745  df-addcfn 5747  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-nc 6102

This theorem is referenced by:  lecadd2  6267