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Theorem lcmftp 14688
 Description: The least common multiple of a triple of integers is the least common multiple of the third integer and the the least common multiple of the first two integers. Although there would be a shorter proof using lcmfunsn 14696, this explicit proof (not based on induction) should be kept. (Proof modification is discouraged.) (Contributed by AV, 23-Aug-2020.)
Assertion
Ref Expression
lcmftp lcm lcm lcm

Proof of Theorem lcmftp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0z 10972 . . . . . . 7
2 eltpg 4005 . . . . . . 7
31, 2ax-mp 5 . . . . . 6
43biimpri 211 . . . . 5
5 tpssi 4130 . . . . 5
64, 5anim12ci 577 . . . 4
7 lcmf0val 14671 . . . 4 lcm
86, 7syl 17 . . 3 lcm
9 0zd 10973 . . . . . . . . . 10
10 lcmcom 14636 . . . . . . . . . 10 lcm lcm
119, 10mpancom 682 . . . . . . . . 9 lcm lcm
12 lcm0val 14637 . . . . . . . . 9 lcm
1311, 12eqtrd 2505 . . . . . . . 8 lcm
1413eqcomd 2477 . . . . . . 7 lcm
15143ad2ant3 1053 . . . . . 6 lcm
1615adantl 473 . . . . 5 lcm
17 0zd 10973 . . . . . . . . . . 11
18 lcmcom 14636 . . . . . . . . . . 11 lcm lcm
1917, 18mpancom 682 . . . . . . . . . 10 lcm lcm
20 lcm0val 14637 . . . . . . . . . 10 lcm
2119, 20eqtrd 2505 . . . . . . . . 9 lcm
2221eqcomd 2477 . . . . . . . 8 lcm
23223ad2ant2 1052 . . . . . . 7 lcm
2423adantl 473 . . . . . 6 lcm
2524oveq1d 6323 . . . . 5 lcm lcm lcm
26 oveq1 6315 . . . . . . 7 lcm lcm
2726oveq1d 6323 . . . . . 6 lcm lcm lcm lcm
2827adantr 472 . . . . 5 lcm lcm lcm lcm
2916, 25, 283eqtrd 2509 . . . 4 lcm lcm
30 lcm0val 14637 . . . . . . . . 9 lcm
3130eqcomd 2477 . . . . . . . 8 lcm
32313ad2ant1 1051 . . . . . . 7 lcm
3332adantl 473 . . . . . 6 lcm
3433oveq1d 6323 . . . . 5 lcm lcm lcm
35133ad2ant3 1053 . . . . . 6 lcm
3635adantl 473 . . . . 5 lcm
37 oveq2 6316 . . . . . . 7 lcm lcm
3837adantr 472 . . . . . 6 lcm lcm
3938oveq1d 6323 . . . . 5 lcm lcm lcm lcm
4034, 36, 393eqtr3d 2513 . . . 4 lcm lcm
41 lcmcl 14645 . . . . . . . 8 lcm
4241nn0zd 11061 . . . . . . 7 lcm
43 lcm0val 14637 . . . . . . . 8 lcm lcm lcm
4443eqcomd 2477 . . . . . . 7 lcm lcm lcm
4542, 44syl 17 . . . . . 6 lcm lcm
46453adant3 1050 . . . . 5 lcm lcm
47 oveq2 6316 . . . . 5 lcm lcm lcm lcm
4846, 47sylan9eqr 2527 . . . 4 lcm lcm
4929, 40, 483jaoian 1359 . . 3 lcm lcm
508, 49eqtrd 2505 . 2 lcm lcm lcm
51423adant3 1050 . . . . . . . . . 10 lcm
52 simp3 1032 . . . . . . . . . 10
5351, 52jca 541 . . . . . . . . 9 lcm
5453adantl 473 . . . . . . . 8 lcm
55 dvdslcm 14642 . . . . . . . 8 lcm lcm lcm lcm lcm lcm
5654, 55syl 17 . . . . . . 7 lcm lcm lcm lcm lcm
57 dvdslcm 14642 . . . . . . . . . . . . . 14 lcm lcm
58573adant3 1050 . . . . . . . . . . . . 13 lcm lcm
59 simp1 1030 . . . . . . . . . . . . . . . . . 18
60 lcmcl 14645 . . . . . . . . . . . . . . . . . . . 20 lcm lcm lcm
6153, 60syl 17 . . . . . . . . . . . . . . . . . . 19 lcm lcm
6261nn0zd 11061 . . . . . . . . . . . . . . . . . 18 lcm lcm
6359, 51, 623jca 1210 . . . . . . . . . . . . . . . . 17 lcm lcm lcm
64 dvdstr 14414 . . . . . . . . . . . . . . . . 17 lcm lcm lcm lcm lcm lcm lcm lcm lcm
6563, 64syl 17 . . . . . . . . . . . . . . . 16 lcm lcm lcm lcm lcm lcm
6665expd 443 . . . . . . . . . . . . . . 15 lcm lcm lcm lcm lcm lcm
6766com12 31 . . . . . . . . . . . . . 14 lcm lcm lcm lcm lcm lcm
6867adantr 472 . . . . . . . . . . . . 13 lcm lcm lcm lcm lcm lcm lcm
6958, 68mpcom 36 . . . . . . . . . . . 12 lcm lcm lcm lcm lcm
7069adantl 473 . . . . . . . . . . 11 lcm lcm lcm lcm lcm
7170com12 31 . . . . . . . . . 10 lcm lcm lcm lcm lcm
7271adantr 472 . . . . . . . . 9 lcm lcm lcm lcm lcm lcm lcm
7372impcom 437 . . . . . . . 8 lcm lcm lcm lcm lcm lcm lcm
74 simpr 468 . . . . . . . . . . . . . . 15 lcm lcm lcm
7557, 74syl 17 . . . . . . . . . . . . . 14 lcm
76753adant3 1050 . . . . . . . . . . . . 13 lcm
7776adantl 473 . . . . . . . . . . . 12 lcm
78 simp2 1031 . . . . . . . . . . . . . . 15
7978, 51, 623jca 1210 . . . . . . . . . . . . . 14 lcm lcm lcm
8079adantl 473 . . . . . . . . . . . . 13 lcm lcm lcm
81 dvdstr 14414 . . . . . . . . . . . . 13 lcm lcm lcm lcm lcm lcm lcm lcm lcm
8280, 81syl 17 . . . . . . . . . . . 12 lcm lcm lcm lcm lcm lcm
8377, 82mpand 689 . . . . . . . . . . 11 lcm lcm lcm lcm lcm
8483com12 31 . . . . . . . . . 10 lcm lcm lcm lcm lcm
8584adantr 472 . . . . . . . . 9 lcm lcm lcm lcm lcm lcm lcm
8685impcom 437 . . . . . . . 8 lcm lcm lcm lcm lcm lcm lcm
87 simpr 468 . . . . . . . . 9 lcm lcm lcm lcm lcm lcm lcm
8887adantl 473 . . . . . . . 8 lcm lcm lcm lcm lcm lcm lcm
8973, 86, 883jca 1210 . . . . . . 7 lcm lcm lcm lcm lcm lcm lcm lcm lcm lcm lcm
9056, 89mpdan 681 . . . . . 6 lcm lcm lcm lcm lcm lcm
91 breq1 4398 . . . . . . . 8 lcm lcm lcm lcm
92 breq1 4398 . . . . . . . 8 lcm lcm lcm lcm
93 breq1 4398 . . . . . . . 8 lcm lcm lcm lcm
9491, 92, 93raltpg 4014 . . . . . . 7 lcm lcm lcm lcm lcm lcm lcm lcm
9594adantl 473 . . . . . 6 lcm lcm lcm lcm lcm lcm lcm lcm
9690, 95mpbird 240 . . . . 5 lcm lcm
97 breq1 4398 . . . . . . . . 9
98 breq1 4398 . . . . . . . . 9
99 breq1 4398 . . . . . . . . 9
10097, 98, 99raltpg 4014 . . . . . . . 8
101100ad2antlr 741 . . . . . . 7
102 simpr 468 . . . . . . . . . . 11
10351ad2antlr 741 . . . . . . . . . . 11 lcm
10452ad2antlr 741 . . . . . . . . . . 11
105102, 103, 1043jca 1210 . . . . . . . . . 10 lcm
106105adantr 472 . . . . . . . . 9 lcm
107 3ioran 1025 . . . . . . . . . . . . . . . . 17
108 eqcom 2478 . . . . . . . . . . . . . . . . . . . . . 22
109108notbii 303 . . . . . . . . . . . . . . . . . . . . 21
110 eqcom 2478 . . . . . . . . . . . . . . . . . . . . . 22
111110notbii 303 . . . . . . . . . . . . . . . . . . . . 21
112109, 111anbi12i 711 . . . . . . . . . . . . . . . . . . . 20
113112biimpi 199 . . . . . . . . . . . . . . . . . . 19
114 ioran 498 . . . . . . . . . . . . . . . . . . 19
115113, 114sylibr 217 . . . . . . . . . . . . . . . . . 18
1161153adant3 1050 . . . . . . . . . . . . . . . . 17
117107, 116sylbi 200 . . . . . . . . . . . . . . . 16
118 id 22 . . . . . . . . . . . . . . . . 17
1191183adant3 1050 . . . . . . . . . . . . . . . 16
120117, 119anim12ci 577 . . . . . . . . . . . . . . 15
121 lcmn0cl 14641 . . . . . . . . . . . . . . 15 lcm
122120, 121syl 17 . . . . . . . . . . . . . 14 lcm
123 nnne0 10664 . . . . . . . . . . . . . . 15 lcm lcm
124123neneqd 2648 . . . . . . . . . . . . . 14 lcm lcm
125122, 124syl 17 . . . . . . . . . . . . 13 lcm
126 eqcom 2478 . . . . . . . . . . . . . . . . . 18
127126notbii 303 . . . . . . . . . . . . . . . . 17
128127biimpi 199 . . . . . . . . . . . . . . . 16
1291283ad2ant3 1053 . . . . . . . . . . . . . . 15
130107, 129sylbi 200 . . . . . . . . . . . . . 14
131130adantr 472 . . . . . . . . . . . . 13
132125, 131jca 541 . . . . . . . . . . . 12 lcm
133132adantr 472 . . . . . . . . . . 11 lcm
134133adantr 472 . . . . . . . . . 10 lcm
135 ioran 498 . . . . . . . . . 10 lcm lcm
136134, 135sylibr 217 . . . . . . . . 9 lcm
137119adantl 473 . . . . . . . . . . . . . . 15
138 nnz 10983 . . . . . . . . . . . . . . 15
139137, 138anim12ci 577 . . . . . . . . . . . . . 14
140 3anass 1011 . . . . . . . . . . . . . 14
141139, 140sylibr 217 . . . . . . . . . . . . 13
142 lcmdvds 14652 . . . . . . . . . . . . 13 lcm
143141, 142syl 17 . . . . . . . . . . . 12 lcm
144143com12 31 . . . . . . . . . . 11 lcm
1451443adant3 1050 . . . . . . . . . 10 lcm
146145impcom 437 . . . . . . . . 9 lcm
147 simp3 1032 . . . . . . . . . 10
148147adantl 473 . . . . . . . . 9
149 lcmledvds 14643 . . . . . . . . . 10 lcm lcm lcm lcm lcm
150149imp 436 . . . . . . . . 9 lcm lcm lcm lcm lcm
151106, 136, 146, 148, 150syl22anc 1293 . . . . . . . 8 lcm lcm
152151ex 441 . . . . . . 7 lcm lcm
153101, 152sylbid 223 . . . . . 6 lcm lcm
154153ralrimiva 2809 . . . . 5 lcm lcm
15596, 154jca 541 . . . 4 lcm lcm lcm lcm
156109biimpi 199 . . . . . . . . . . . . . . . 16
157111biimpi 199 . . . . . . . . . . . . . . . 16
158156, 157anim12i 576 . . . . . . . . . . . . . . 15
159158, 114sylibr 217 . . . . . . . . . . . . . 14
1601593adant3 1050 . . . . . . . . . . . . 13
161107, 160sylbi 200 . . . . . . . . . . . 12
162161, 119anim12ci 577 . . . . . . . . . . 11
163162, 121syl 17 . . . . . . . . . 10 lcm
164163, 124syl 17 . . . . . . . . 9 lcm
165164, 131jca 541 . . . . . . . 8 lcm
166165, 135sylibr 217 . . . . . . 7 lcm
16754, 166jca 541 . . . . . 6 lcm lcm
168 lcmn0cl 14641 . . . . . 6 lcm lcm lcm lcm
169167, 168syl 17 . . . . 5 lcm lcm
1705adantl 473 . . . . 5
171 tpfi 7865 . . . . . 6
172171a1i 11 . . . . 5
1733a1i 11 . . . . . . . . 9
174173biimpd 212 . . . . . . . 8
175174con3d 140 . . . . . . 7
176175impcom 437 . . . . . 6
177 df-nel 2644 . . . . . 6
178176, 177sylibr 217 . . . . 5
179 lcmf 14685 . . . . 5 lcm lcm lcm lcm lcm lcm lcm lcm lcm
180169, 170, 172, 178, 179syl13anc 1294 . . . 4 lcm lcm lcm lcm lcm lcm lcm
181155, 180mpbird 240 . . 3 lcm lcm lcm
182181eqcomd 2477 . 2 lcm lcm lcm
18350, 182pm2.61ian 807 1 lcm lcm lcm
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wo 375   wa 376   w3o 1006   w3a 1007   wceq 1452   wcel 1904   wnel 2642  wral 2756   wss 3390  ctp 3963   class class class wbr 4395  cfv 5589  (class class class)co 6308  cfn 7587  cc0 9557   cle 9694  cn 10631  cn0 10893  cz 10961   cdvds 14382   lcm clcm 14626  lcmclcmf 14627 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-prod 14037  df-dvds 14383  df-gcd 14548  df-lcm 14630  df-lcmf 14632 This theorem is referenced by:  lcmf2a3a4e12  14699
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