Theorem lecadd2 6266

Description: Cardinal addition preserves cardinal less than. Biconditional form of corollary 4 of theorem XI.3.2 of [Rosser] p. 391. (Contributed by Scott Fenton, 2-Aug-2019.)

Assertion

Ref	Expression
lecadd2	⊢ ((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) → ((M +c N) ≤c (M +c P) ↔ N ≤c P))

Proof of Theorem lecadd2

Dummy variable q is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnc 6146	. . . . . 6 ⊢ (M ∈ Nn → M ∈ NC )
2		ncaddccl 6144	. . . . . 6 ⊢ ((M ∈ NC ∧ N ∈ NC ) → (M +c N) ∈ NC )
3	1, 2	sylan 457	. . . . 5 ⊢ ((M ∈ Nn ∧ N ∈ NC ) → (M +c N) ∈ NC )
4	3	3adant3 975	. . . 4 ⊢ ((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) → (M +c N) ∈ NC )
5		ncaddccl 6144	. . . . . 6 ⊢ ((M ∈ NC ∧ P ∈ NC ) → (M +c P) ∈ NC )
6	1, 5	sylan 457	. . . . 5 ⊢ ((M ∈ Nn ∧ P ∈ NC ) → (M +c P) ∈ NC )
7	6	3adant2 974	. . . 4 ⊢ ((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) → (M +c P) ∈ NC )
8		dflec2 6210	. . . 4 ⊢ (((M +c N) ∈ NC ∧ (M +c P) ∈ NC ) → ((M +c N) ≤c (M +c P) ↔ ∃q ∈ NC (M +c P) = ((M +c N) +c q)))
9	4, 7, 8	syl2anc 642	. . 3 ⊢ ((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) → ((M +c N) ≤c (M +c P) ↔ ∃q ∈ NC (M +c P) = ((M +c N) +c q)))
10		addcass 4415	. . . . . 6 ⊢ ((M +c N) +c q) = (M +c (N +c q))
11	10	eqeq2i 2363	. . . . 5 ⊢ ((M +c P) = ((M +c N) +c q) ↔ (M +c P) = (M +c (N +c q)))
12		simpl1 958	. . . . . . 7 ⊢ (((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) ∧ q ∈ NC ) → M ∈ Nn )
13		simpl3 960	. . . . . . 7 ⊢ (((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) ∧ q ∈ NC ) → P ∈ NC )
14		ncaddccl 6144	. . . . . . . 8 ⊢ ((N ∈ NC ∧ q ∈ NC ) → (N +c q) ∈ NC )
15	14	3ad2antl2 1118	. . . . . . 7 ⊢ (((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) ∧ q ∈ NC ) → (N +c q) ∈ NC )
16		addccan2nc 6265	. . . . . . 7 ⊢ ((M ∈ Nn ∧ P ∈ NC ∧ (N +c q) ∈ NC ) → ((M +c P) = (M +c (N +c q)) ↔ P = (N +c q)))
17	12, 13, 15, 16	syl3anc 1182	. . . . . 6 ⊢ (((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) ∧ q ∈ NC ) → ((M +c P) = (M +c (N +c q)) ↔ P = (N +c q)))
18		addlecncs 6209	. . . . . . . 8 ⊢ ((N ∈ NC ∧ q ∈ NC ) → N ≤c (N +c q))
19	18	3ad2antl2 1118	. . . . . . 7 ⊢ (((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) ∧ q ∈ NC ) → N ≤c (N +c q))
20		breq2 4643	. . . . . . 7 ⊢ (P = (N +c q) → (N ≤c P ↔ N ≤c (N +c q)))
21	19, 20	syl5ibrcom 213	. . . . . 6 ⊢ (((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) ∧ q ∈ NC ) → (P = (N +c q) → N ≤c P))
22	17, 21	sylbid 206	. . . . 5 ⊢ (((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) ∧ q ∈ NC ) → ((M +c P) = (M +c (N +c q)) → N ≤c P))
23	11, 22	syl5bi 208	. . . 4 ⊢ (((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) ∧ q ∈ NC ) → ((M +c P) = ((M +c N) +c q) → N ≤c P))
24	23	rexlimdva 2738	. . 3 ⊢ ((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) → (∃q ∈ NC (M +c P) = ((M +c N) +c q) → N ≤c P))
25	9, 24	sylbid 206	. 2 ⊢ ((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) → ((M +c N) ≤c (M +c P) → N ≤c P))
26		leaddc2 6215	. . . 4 ⊢ (((M ∈ NC ∧ N ∈ NC ∧ P ∈ NC ) ∧ N ≤c P) → (M +c N) ≤c (M +c P))
27	26	ex 423	. . 3 ⊢ ((M ∈ NC ∧ N ∈ NC ∧ P ∈ NC ) → (N ≤c P → (M +c N) ≤c (M +c P)))
28	1, 27	syl3an1 1215	. 2 ⊢ ((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) → (N ≤c P → (M +c N) ≤c (M +c P)))
29	25, 28	impbid 183	1 ⊢ ((M ∈ Nn ∧ N ∈ NC ∧ P ∈ NC ) → ((M +c N) ≤c (M +c P) ↔ N ≤c P))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   Nn cnnc 4373   +_c cplc 4375   class class class wbr 4639   NC cncs 6088   ≤_c clec 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 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-f1 4792  df-fo 4793  df-f1o 4794  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-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-lec 6099  df-nc 6101

This theorem is referenced by: (None)