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Theorem sgnsv 30500
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 3426 . . 3 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6496 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 sgnsval.b . . . . . 6 𝐵 = (Base‘𝑅)
53, 4syl6eqr 2825 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6 fveq2 6496 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
7 sgnsval.0 . . . . . . . . 9 0 = (0g𝑅)
86, 7syl6eqr 2825 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
98adantr 473 . . . . . . 7 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (0g𝑟) = 0 )
109eqeq2d 2781 . . . . . 6 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (𝑥 = (0g𝑟) ↔ 𝑥 = 0 ))
11 fveq2 6496 . . . . . . . . . 10 (𝑟 = 𝑅 → (lt‘𝑟) = (lt‘𝑅))
12 sgnsval.l . . . . . . . . . 10 < = (lt‘𝑅)
1311, 12syl6eqr 2825 . . . . . . . . 9 (𝑟 = 𝑅 → (lt‘𝑟) = < )
1413adantr 473 . . . . . . . 8 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (lt‘𝑟) = < )
15 eqidd 2772 . . . . . . . 8 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → 𝑥 = 𝑥)
169, 14, 15breq123d 4939 . . . . . . 7 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → ((0g𝑟)(lt‘𝑟)𝑥0 < 𝑥))
1716ifbid 4366 . . . . . 6 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → if((0g𝑟)(lt‘𝑟)𝑥, 1, -1) = if( 0 < 𝑥, 1, -1))
1810, 17ifbieq2d 4369 . . . . 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 5007 . . . 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 30499 . . . 4 sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1))))
2119, 20, 4mptfvmpt 6814 . . 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, 22syl5eq 2819 1 (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wcel 2051  Vcvv 3408  ifcif 4344   class class class wbr 4925  cmpt 5004  cfv 6185  0cc0 10333  1c1 10334  -cneg 10669  Basecbs 16337  0gc0g 16567  ltcplt 17421  sgnscsgns 30498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-sgns 30499
This theorem is referenced by:  sgnsval  30501  sgnsf  30502
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