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Theorem tz7.44-1 7091
Description: The value of at . Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypotheses
Ref Expression
tz7.44.1
tz7.44.2
tz7.44-1.3
Assertion
Ref Expression
tz7.44-1
Distinct variable groups:   ,   , ,   ,   ,   ,

Proof of Theorem tz7.44-1
StepHypRef Expression
1 fveq2 5871 . . . 4
2 reseq2 5273 . . . . . 6
3 res0 5283 . . . . . 6
42, 3syl6eq 2514 . . . . 5
54fveq2d 5875 . . . 4
61, 5eqeq12d 2479 . . 3
7 tz7.44.2 . . 3
86, 7vtoclga 3173 . 2
9 0ex 4582 . . 3
10 iftrue 3947 . . . 4
11 tz7.44.1 . . . 4
12 tz7.44-1.3 . . . 4
1310, 11, 12fvmpt 5956 . . 3
149, 13ax-mp 5 . 2
158, 14syl6eq 2514 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818   cvv 3109   c0 3784  ifcif 3941  U.cuni 4249  e.cmpt 4510  Limwlim 4884  domcdm 5004  rancrn 5005  |`cres 5006  `cfv 5593
This theorem is referenced by:  rdg0  7106
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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
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-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  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-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-res 5016  df-iota 5556  df-fun 5595  df-fv 5601
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