Theorem lemuc1 6253

Description: Multiplication law for cardinal less than. Theorem XI.2.35 of [Rosser] p. 380. (Contributed by Scott Fenton, 31-Jul-2019.)

Assertion

Ref	Expression
lemuc1	⊢ (((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ∧ A ≤c B) → (A ·c C) ≤c (B ·c C))

Proof of Theorem lemuc1

Dummy variable q is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dflec2 6210	. . . 4 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (A ≤c B ↔ ∃q ∈ NC B = (A +c q)))
2	1	3adant3 975	. . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → (A ≤c B ↔ ∃q ∈ NC B = (A +c q)))
3		muccl 6132	. . . . . . . . 9 ⊢ ((A ∈ NC ∧ C ∈ NC ) → (A ·c C) ∈ NC )
4	3	adantr 451	. . . . . . . 8 ⊢ (((A ∈ NC ∧ C ∈ NC ) ∧ q ∈ NC ) → (A ·c C) ∈ NC )
5		muccl 6132	. . . . . . . . . 10 ⊢ ((q ∈ NC ∧ C ∈ NC ) → (q ·c C) ∈ NC )
6	5	ancoms 439	. . . . . . . . 9 ⊢ ((C ∈ NC ∧ q ∈ NC ) → (q ·c C) ∈ NC )
7	6	adantll 694	. . . . . . . 8 ⊢ (((A ∈ NC ∧ C ∈ NC ) ∧ q ∈ NC ) → (q ·c C) ∈ NC )
8		addlecncs 6209	. . . . . . . 8 ⊢ (((A ·c C) ∈ NC ∧ (q ·c C) ∈ NC ) → (A ·c C) ≤c ((A ·c C) +c (q ·c C)))
9	4, 7, 8	syl2anc 642	. . . . . . 7 ⊢ (((A ∈ NC ∧ C ∈ NC ) ∧ q ∈ NC ) → (A ·c C) ≤c ((A ·c C) +c (q ·c C)))
10		simpll 730	. . . . . . . 8 ⊢ (((A ∈ NC ∧ C ∈ NC ) ∧ q ∈ NC ) → A ∈ NC )
11		simpr 447	. . . . . . . 8 ⊢ (((A ∈ NC ∧ C ∈ NC ) ∧ q ∈ NC ) → q ∈ NC )
12		simplr 731	. . . . . . . 8 ⊢ (((A ∈ NC ∧ C ∈ NC ) ∧ q ∈ NC ) → C ∈ NC )
13		addcdir 6251	. . . . . . . 8 ⊢ ((A ∈ NC ∧ q ∈ NC ∧ C ∈ NC ) → ((A +c q) ·c C) = ((A ·c C) +c (q ·c C)))
14	10, 11, 12, 13	syl3anc 1182	. . . . . . 7 ⊢ (((A ∈ NC ∧ C ∈ NC ) ∧ q ∈ NC ) → ((A +c q) ·c C) = ((A ·c C) +c (q ·c C)))
15	9, 14	breqtrrd 4665	. . . . . 6 ⊢ (((A ∈ NC ∧ C ∈ NC ) ∧ q ∈ NC ) → (A ·c C) ≤c ((A +c q) ·c C))
16		oveq1 5530	. . . . . . 7 ⊢ (B = (A +c q) → (B ·c C) = ((A +c q) ·c C))
17	16	breq2d 4651	. . . . . 6 ⊢ (B = (A +c q) → ((A ·c C) ≤c (B ·c C) ↔ (A ·c C) ≤c ((A +c q) ·c C)))
18	15, 17	syl5ibrcom 213	. . . . 5 ⊢ (((A ∈ NC ∧ C ∈ NC ) ∧ q ∈ NC ) → (B = (A +c q) → (A ·c C) ≤c (B ·c C)))
19	18	rexlimdva 2738	. . . 4 ⊢ ((A ∈ NC ∧ C ∈ NC ) → (∃q ∈ NC B = (A +c q) → (A ·c C) ≤c (B ·c C)))
20	19	3adant2 974	. . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → (∃q ∈ NC B = (A +c q) → (A ·c C) ≤c (B ·c C)))
21	2, 20	sylbid 206	. 2 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → (A ≤c B → (A ·c C) ≤c (B ·c C)))
22	21	imp 418	1 ⊢ (((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ∧ A ≤c B) → (A ·c C) ≤c (B ·c C))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   +_c cplc 4375   class class class wbr 4639  (class class class)co 5525   NC cncs 6088   ≤_c clec 6089   ·_c cmuc 6092

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-pprod 5738  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-cross 5764  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  df-muc 6102

This theorem is referenced by:  lemuc2  6254