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Theorem seqeq3 12112
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq3

Proof of Theorem seqeq3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 5870 . . . . . . 7
21oveq2d 6312 . . . . . 6
32opeq2d 4224 . . . . 5
43mpt2eq3dv 6363 . . . 4
5 fveq1 5870 . . . . 5
65opeq2d 4224 . . . 4
7 rdgeq12 7098 . . . 4
84, 6, 7syl2anc 661 . . 3
98imaeq1d 5341 . 2
10 df-seq 12108 . 2
11 df-seq 12108 . 2
129, 10, 113eqtr4g 2523 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  =wceq 1395   cvv 3109  <.cop 4035  "cima 5007  `cfv 5593  (class class class)co 6296  e.cmpt2 6298   com 6700  reccrdg 7094  1c1 9514   caddc 9516  seqcseq 12107
This theorem is referenced by:  seqeq3d  12115  cbvprod  13722  iprodmul  13796  geolim3  22735  leibpilem2  23272  basel  23363  faclim  29171  ovoliunnfl  30056  voliunnfl  30058  heiborlem10  30316  binomcxplemnn0  31254  binomcxplemdvsum  31260  binomcxp  31262  fourierdlem112  32001  fouriersw  32014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-recs 7061  df-rdg 7095  df-seq 12108
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