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Theorem lcmfunsnlem 14693
 Description: Lemma for lcmfdvds 14694 and lcmfunsn 14696. These two theorems must be proven simultaneously by induction on the cardinality of a finite set , because they depend on each other. This can be seen by the two parts lcmfunsnlem1 14689 and lcmfunsnlem2 14692 of the induction step, each of them using both induction hypotheses. (Contributed by AV, 26-Aug-2020.)
Assertion
Ref Expression
lcmfunsnlem lcm lcm lcm lcm
Distinct variable group:   ,,,

Proof of Theorem lcmfunsnlem
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 3439 . . . 4
2 raleq 2973 . . . . . . 7
3 fveq2 5879 . . . . . . . 8 lcm lcm
43breq1d 4405 . . . . . . 7 lcm lcm
52, 4imbi12d 327 . . . . . 6 lcm lcm
65ralbidv 2829 . . . . 5 lcm lcm
7 uneq1 3572 . . . . . . . 8
87fveq2d 5883 . . . . . . 7 lcm lcm
93oveq1d 6323 . . . . . . 7 lcm lcm lcm lcm
108, 9eqeq12d 2486 . . . . . 6 lcm lcm lcm lcm lcm lcm
1110ralbidv 2829 . . . . 5 lcm lcm lcm lcm lcm lcm
126, 11anbi12d 725 . . . 4 lcm lcm lcm lcm lcm lcm lcm lcm
131, 12imbi12d 327 . . 3 lcm lcm lcm lcm lcm lcm lcm lcm
14 sseq1 3439 . . . 4
15 raleq 2973 . . . . . . 7
16 fveq2 5879 . . . . . . . 8 lcm lcm
1716breq1d 4405 . . . . . . 7 lcm lcm
1815, 17imbi12d 327 . . . . . 6 lcm lcm
1918ralbidv 2829 . . . . 5 lcm lcm
20 uneq1 3572 . . . . . . . 8
2120fveq2d 5883 . . . . . . 7 lcm lcm
2216oveq1d 6323 . . . . . . 7 lcm lcm lcm lcm
2321, 22eqeq12d 2486 . . . . . 6 lcm lcm lcm lcm lcm lcm
2423ralbidv 2829 . . . . 5 lcm lcm lcm lcm lcm lcm
2519, 24anbi12d 725 . . . 4 lcm lcm lcm lcm lcm lcm lcm lcm
2614, 25imbi12d 327 . . 3 lcm lcm lcm lcm lcm lcm lcm lcm
27 sseq1 3439 . . . 4
28 raleq 2973 . . . . . . 7
29 fveq2 5879 . . . . . . . 8 lcm lcm
3029breq1d 4405 . . . . . . 7 lcm lcm
3128, 30imbi12d 327 . . . . . 6 lcm lcm
3231ralbidv 2829 . . . . 5 lcm lcm
33 uneq1 3572 . . . . . . . 8
3433fveq2d 5883 . . . . . . 7 lcm lcm
3529oveq1d 6323 . . . . . . 7 lcm lcm lcm lcm
3634, 35eqeq12d 2486 . . . . . 6 lcm lcm lcm lcm lcm lcm
3736ralbidv 2829 . . . . 5 lcm lcm lcm lcm lcm lcm
3832, 37anbi12d 725 . . . 4 lcm lcm lcm lcm lcm lcm lcm lcm
3927, 38imbi12d 327 . . 3 lcm lcm lcm lcm lcm lcm lcm lcm
40 sseq1 3439 . . . 4
41 raleq 2973 . . . . . . 7
42 fveq2 5879 . . . . . . . 8 lcm lcm
4342breq1d 4405 . . . . . . 7 lcm lcm
4441, 43imbi12d 327 . . . . . 6 lcm lcm
4544ralbidv 2829 . . . . 5 lcm lcm
46 uneq1 3572 . . . . . . . 8
4746fveq2d 5883 . . . . . . 7 lcm lcm
4842oveq1d 6323 . . . . . . 7 lcm lcm lcm lcm
4947, 48eqeq12d 2486 . . . . . 6 lcm lcm lcm lcm lcm lcm
5049ralbidv 2829 . . . . 5 lcm lcm lcm lcm lcm lcm
5145, 50anbi12d 725 . . . 4 lcm lcm lcm lcm lcm lcm lcm lcm
5240, 51imbi12d 327 . . 3 lcm lcm lcm lcm lcm lcm lcm lcm
53 lcmf0 14686 . . . . . . . 8 lcm
54 1dvds 14394 . . . . . . . 8
5553, 54syl5eqbr 4429 . . . . . . 7 lcm
5655a1d 25 . . . . . 6 lcm
5756adantl 473 . . . . 5 lcm
5857ralrimiva 2809 . . . 4 lcm
59 uncom 3569 . . . . . . . . . 10
60 un0 3762 . . . . . . . . . 10
6159, 60eqtri 2493 . . . . . . . . 9
6261a1i 11 . . . . . . . 8
6362fveq2d 5883 . . . . . . 7 lcm lcm
64 lcmfsn 14687 . . . . . . 7 lcm
6553a1i 11 . . . . . . . . 9 lcm
6665oveq1d 6323 . . . . . . . 8 lcm lcm lcm
67 1z 10991 . . . . . . . . 9
68 lcmcom 14636 . . . . . . . . 9 lcm lcm
6967, 68mpan 684 . . . . . . . 8 lcm lcm
70 lcm1 14654 . . . . . . . 8 lcm
7166, 69, 703eqtrrd 2510 . . . . . . 7 lcm lcm
7263, 64, 713eqtrd 2509 . . . . . 6 lcm lcm lcm
7372adantl 473 . . . . 5 lcm lcm lcm
7473ralrimiva 2809 . . . 4 lcm lcm lcm
7558, 74jca 541 . . 3 lcm lcm lcm lcm
76 unss 3599 . . . . . . . 8
77 simpl 464 . . . . . . . 8
7876, 77sylbir 218 . . . . . . 7
7978adantl 473 . . . . . 6
80 vex 3034 . . . . . . . . . . 11
8180snss 4087 . . . . . . . . . 10
82 lcmfunsnlem1 14689 . . . . . . . . . . . 12 lcm lcm lcm lcm lcm
83 lcmfunsnlem2 14692 . . . . . . . . . . . 12 lcm lcm lcm lcm lcm lcm lcm
8482, 83jca 541 . . . . . . . . . . 11 lcm lcm lcm lcm lcm lcm lcm lcm
85843exp1 1249 . . . . . . . . . 10 lcm lcm lcm lcm lcm lcm lcm lcm
8681, 85sylbir 218 . . . . . . . . 9 lcm lcm lcm lcm lcm lcm lcm lcm
8786impcom 437 . . . . . . . 8 lcm lcm lcm lcm lcm lcm lcm lcm
8876, 87sylbir 218 . . . . . . 7 lcm lcm lcm lcm lcm lcm lcm lcm
8988impcom 437 . . . . . 6 lcm lcm lcm lcm lcm lcm lcm lcm
9079, 89embantd 55 . . . . 5 lcm lcm lcm lcm lcm lcm lcm lcm
9190ex 441 . . . 4 lcm lcm lcm lcm lcm lcm lcm lcm
9291com23 80 . . 3 lcm lcm lcm lcm lcm lcm lcm lcm
9313, 26, 39, 52, 75, 92findcard2 7829 . 2 lcm lcm lcm lcm
9493impcom 437 1 lcm lcm lcm lcm
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   w3a 1007   wceq 1452   wcel 1904  wral 2756   cun 3388   wss 3390  c0 3722  csn 3959   class class class wbr 4395  cfv 5589  (class class class)co 6308  cfn 7587  c1 9558  cz 10961  cabs 13374   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:  lcmfdvds  14694  lcmfunsn  14696
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