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Mirrors > Home > NFE Home > Th. List > lemuc1 | Unicode 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 | NC NC NC c ·c c ·c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dflec2 6211 | . . . 4 NC NC c NC | |
2 | 1 | 3adant3 975 | . . 3 NC NC NC c NC |
3 | muccl 6133 | . . . . . . . . 9 NC NC ·c NC | |
4 | 3 | adantr 451 | . . . . . . . 8 NC NC NC ·c NC |
5 | muccl 6133 | . . . . . . . . . 10 NC NC ·c NC | |
6 | 5 | ancoms 439 | . . . . . . . . 9 NC NC ·c NC |
7 | 6 | adantll 694 | . . . . . . . 8 NC NC NC ·c NC |
8 | addlecncs 6210 | . . . . . . . 8 ·c NC ·c NC ·c c ·c ·c | |
9 | 4, 7, 8 | syl2anc 642 | . . . . . . 7 NC NC NC ·c c ·c ·c |
10 | simpll 730 | . . . . . . . 8 NC NC NC NC | |
11 | simpr 447 | . . . . . . . 8 NC NC NC NC | |
12 | simplr 731 | . . . . . . . 8 NC NC NC NC | |
13 | addcdir 6252 | . . . . . . . 8 NC NC NC ·c ·c ·c | |
14 | 10, 11, 12, 13 | syl3anc 1182 | . . . . . . 7 NC NC NC ·c ·c ·c |
15 | 9, 14 | breqtrrd 4666 | . . . . . 6 NC NC NC ·c c ·c |
16 | oveq1 5531 | . . . . . . 7 ·c ·c | |
17 | 16 | breq2d 4652 | . . . . . 6 ·c c ·c ·c c ·c |
18 | 15, 17 | syl5ibrcom 213 | . . . . 5 NC NC NC ·c c ·c |
19 | 18 | rexlimdva 2739 | . . . 4 NC NC NC ·c c ·c |
20 | 19 | 3adant2 974 | . . 3 NC NC NC NC ·c c ·c |
21 | 2, 20 | sylbid 206 | . 2 NC NC NC c ·c c ·c |
22 | 21 | imp 418 | 1 NC NC NC c ·c c ·c |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 w3a 934 wceq 1642 wcel 1710 wrex 2616 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 |
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