Theorem lemuc1 6254

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 6211	. . . 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 6133	. . . . . . . . 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 6133	. . . . . . . . . 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 6210	. . . . . . . 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 6252	. . . . . . . 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 4666	. . . . . 6 ⊢ (((A ∈ NC ∧ C ∈ NC ) ∧ q ∈ NC ) → (A ·c C) ≤c ((A +c q) ·c C))
16		oveq1 5531	. . . . . . 7 ⊢ (B = (A +c q) → (B ·c C) = ((A +c q) ·c C))
17	16	breq2d 4652	. . . . . 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 2739	. . . 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 2616   +_c cplc 4376   class class class wbr 4640  (class class class)co 5526   NC cncs 6089   ≤_c clec 6090   ·_c cmuc 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 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-pprod 5739  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-cross 5765  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-lec 6100  df-nc 6102  df-muc 6103

This theorem is referenced by:  lemuc2  6255