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Theorem lcmval 14526
 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 14443 and gcdval 14444. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)
Assertion
Ref Expression
lcmval lcm inf
Distinct variable groups:   ,   ,

Proof of Theorem lcmval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2433 . . . 4
21orbi1d 707 . . 3
3 breq1 4429 . . . . . 6
43anbi1d 709 . . . . 5
54rabbidv 3079 . . . 4
65infeq1d 7999 . . 3 inf inf
72, 6ifbieq2d 3940 . 2 inf inf
8 eqeq1 2433 . . . 4
98orbi2d 706 . . 3
10 breq1 4429 . . . . . 6
1110anbi2d 708 . . . . 5
1211rabbidv 3079 . . . 4
1312infeq1d 7999 . . 3 inf inf
149, 13ifbieq2d 3940 . 2 inf inf
15 df-lcm 14522 . 2 lcm inf
16 c0ex 9636 . . 3
17 ltso 9713 . . . 4
1817infex 8015 . . 3 inf
1916, 18ifex 3983 . 2 inf
207, 14, 15, 19ovmpt2 6446 1 lcm inf
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 369   wa 370   wceq 1437   wcel 1870  crab 2786  cif 3915   class class class wbr 4426  (class class class)co 6305  infcinf 7961  cr 9537  cc0 9538   clt 9674  cn 10609  cz 10937   cdvds 14283   lcm clcm 14518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-mulcl 9600  ax-i2m1 9606  ax-pre-lttri 9612  ax-pre-lttrn 9613 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-inf 7963  df-pnf 9676  df-mnf 9677  df-ltxr 9679  df-lcm 14522 This theorem is referenced by:  lcmcom  14528  lcm0val  14529  lcmn0val  14530  lcmass  14550  lcmfpr  14571
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