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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 ) Ac B) → (A ·c C) ≤c (B ·c C))

Proof of Theorem lemuc1
Dummy variable q is distinct from all other variables.
StepHypRef Expression
1 dflec2 6210 . . . 4 ((A NC B NC ) → (Ac Bq NC B = (A +c q)))
213adant3 975 . . 3 ((A NC B NC C NC ) → (Ac Bq NC B = (A +c q)))
3 muccl 6132 . . . . . . . . 9 ((A NC C NC ) → (A ·c C) NC )
43adantr 451 . . . . . . . 8 (((A NC C NC ) q NC ) → (A ·c C) NC )
5 muccl 6132 . . . . . . . . . 10 ((q NC C NC ) → (q ·c C) NC )
65ancoms 439 . . . . . . . . 9 ((C NC q NC ) → (q ·c C) NC )
76adantll 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)))
94, 7, 8syl2anc 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)))
1410, 11, 12, 13syl3anc 1182 . . . . . . 7 (((A NC C NC ) q NC ) → ((A +c q) ·c C) = ((A ·c C) +c (q ·c C)))
159, 14breqtrrd 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))
1716breq2d 4651 . . . . . 6 (B = (A +c q) → ((A ·c C) ≤c (B ·c C) ↔ (A ·c C) ≤c ((A +c q) ·c C)))
1815, 17syl5ibrcom 213 . . . . 5 (((A NC C NC ) q NC ) → (B = (A +c q) → (A ·c C) ≤c (B ·c C)))
1918rexlimdva 2738 . . . 4 ((A NC C NC ) → (q NC B = (A +c q) → (A ·c C) ≤c (B ·c C)))
20193adant2 974 . . 3 ((A NC B NC C NC ) → (q NC B = (A +c q) → (A ·c C) ≤c (B ·c C)))
212, 20sylbid 206 . 2 ((A NC B NC C NC ) → (Ac B → (A ·c C) ≤c (B ·c C)))
2221imp 418 1 (((A NC B NC C NC ) Ac B) → (A ·c C) ≤c (B ·c C))
 Colors of variables: wff setvar class 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
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