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Theorem sgnsv 33038
Description: The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.)
Hypotheses
Ref Expression
sgnsval.b 𝐵 = (Base‘𝑅)
sgnsval.0 0 = (0g𝑅)
sgnsval.l < = (lt‘𝑅)
sgnsval.s 𝑆 = (sgns𝑅)
Assertion
Ref Expression
sgnsv (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
Distinct variable groups:   𝑥, 0   𝑥, <   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem sgnsv
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 sgnsval.s . 2 𝑆 = (sgns𝑅)
2 elex 3482 . . 3 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6901 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 sgnsval.b . . . . . 6 𝐵 = (Base‘𝑅)
53, 4eqtr4di 2784 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6 fveq2 6901 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
7 sgnsval.0 . . . . . . . . 9 0 = (0g𝑅)
86, 7eqtr4di 2784 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
98adantr 479 . . . . . . 7 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (0g𝑟) = 0 )
109eqeq2d 2737 . . . . . 6 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (𝑥 = (0g𝑟) ↔ 𝑥 = 0 ))
11 fveq2 6901 . . . . . . . . . 10 (𝑟 = 𝑅 → (lt‘𝑟) = (lt‘𝑅))
12 sgnsval.l . . . . . . . . . 10 < = (lt‘𝑅)
1311, 12eqtr4di 2784 . . . . . . . . 9 (𝑟 = 𝑅 → (lt‘𝑟) = < )
1413adantr 479 . . . . . . . 8 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (lt‘𝑟) = < )
15 eqidd 2727 . . . . . . . 8 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → 𝑥 = 𝑥)
169, 14, 15breq123d 5167 . . . . . . 7 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → ((0g𝑟)(lt‘𝑟)𝑥0 < 𝑥))
1716ifbid 4556 . . . . . 6 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → if((0g𝑟)(lt‘𝑟)𝑥, 1, -1) = if( 0 < 𝑥, 1, -1))
1810, 17ifbieq2d 4559 . . . . 5 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1)) = if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))
195, 18mpteq12dva 5242 . . . 4 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1))) = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
20 df-sgns 33037 . . . 4 sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1))))
2119, 20, 4mptfvmpt 7245 . . 3 (𝑅 ∈ V → (sgns𝑅) = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
222, 21syl 17 . 2 (𝑅𝑉 → (sgns𝑅) = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
231, 22eqtrid 2778 1 (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  ifcif 4533   class class class wbr 5153  cmpt 5236  cfv 6554  0cc0 11158  1c1 11159  -cneg 11495  Basecbs 17213  0gc0g 17454  ltcplt 18333  sgnscsgns 33036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-sgns 33037
This theorem is referenced by:  sgnsval  33039  sgnsf  33040
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