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Mirrors > Home > NFE Home > Th. List > lemuc1 | GIF version |
Description: Multiplication law for cardinal less than. Theorem XI.2.35 of [Rosser] p. 380. (Contributed by Scott Fenton, 31-Jul-2019.) |
Ref | Expression |
---|---|
lemuc1 | ⊢ (((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ∧ A ≤c B) → (A ·c C) ≤c (B ·c C)) |
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)) |
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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