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Mirrors > Home > MPE Home > Th. List > lcmval | Structured version Visualization version Unicode version |
Description: Value of the lcm operator. lcm is the least common multiple of and . If either or is , the result is defined conventionally as . Contrast with df-gcd 15217 and gcdval 15218. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.) |
Ref | Expression |
---|---|
lcmval | lcm inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2626 | . . . 4 | |
2 | 1 | orbi1d 739 | . . 3 |
3 | breq1 4656 | . . . . . 6 | |
4 | 3 | anbi1d 741 | . . . . 5 |
5 | 4 | rabbidv 3189 | . . . 4 |
6 | 5 | infeq1d 8383 | . . 3 inf inf |
7 | 2, 6 | ifbieq2d 4111 | . 2 inf inf |
8 | eqeq1 2626 | . . . 4 | |
9 | 8 | orbi2d 738 | . . 3 |
10 | breq1 4656 | . . . . . 6 | |
11 | 10 | anbi2d 740 | . . . . 5 |
12 | 11 | rabbidv 3189 | . . . 4 |
13 | 12 | infeq1d 8383 | . . 3 inf inf |
14 | 9, 13 | ifbieq2d 4111 | . 2 inf inf |
15 | df-lcm 15303 | . 2 lcm inf | |
16 | c0ex 10034 | . . 3 | |
17 | ltso 10118 | . . . 4 | |
18 | 17 | infex 8399 | . . 3 inf |
19 | 16, 18 | ifex 4156 | . 2 inf |
20 | 7, 14, 15, 19 | ovmpt2 6796 | 1 lcm inf |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wo 383 wa 384 wceq 1483 wcel 1990 crab 2916 cif 4086 class class class wbr 4653 (class class class)co 6650 infcinf 8347 cr 9935 cc0 9936 clt 10074 cn 11020 cz 11377 cdvds 14983 lcm clcm 15301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-i2m1 10004 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-lcm 15303 |
This theorem is referenced by: lcmcom 15306 lcm0val 15307 lcmn0val 15308 lcmass 15327 lcmfpr 15340 |
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