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Theorem sgnsv 30729
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 3510 . . 3 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6663 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 sgnsval.b . . . . . 6 𝐵 = (Base‘𝑅)
53, 4syl6eqr 2871 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6 fveq2 6663 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
7 sgnsval.0 . . . . . . . . 9 0 = (0g𝑅)
86, 7syl6eqr 2871 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
98adantr 481 . . . . . . 7 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (0g𝑟) = 0 )
109eqeq2d 2829 . . . . . 6 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (𝑥 = (0g𝑟) ↔ 𝑥 = 0 ))
11 fveq2 6663 . . . . . . . . . 10 (𝑟 = 𝑅 → (lt‘𝑟) = (lt‘𝑅))
12 sgnsval.l . . . . . . . . . 10 < = (lt‘𝑅)
1311, 12syl6eqr 2871 . . . . . . . . 9 (𝑟 = 𝑅 → (lt‘𝑟) = < )
1413adantr 481 . . . . . . . 8 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (lt‘𝑟) = < )
15 eqidd 2819 . . . . . . . 8 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → 𝑥 = 𝑥)
169, 14, 15breq123d 5071 . . . . . . 7 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → ((0g𝑟)(lt‘𝑟)𝑥0 < 𝑥))
1716ifbid 4485 . . . . . 6 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → if((0g𝑟)(lt‘𝑟)𝑥, 1, -1) = if( 0 < 𝑥, 1, -1))
1810, 17ifbieq2d 4488 . . . . 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 5141 . . . 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 30728 . . . 4 sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1))))
2119, 20, 4mptfvmpt 6981 . . 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 2865 1 (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  ifcif 4463   class class class wbr 5057  cmpt 5137  cfv 6348  0cc0 10525  1c1 10526  -cneg 10859  Basecbs 16471  0gc0g 16701  ltcplt 17539  sgnscsgns 30727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-sgns 30728
This theorem is referenced by:  sgnsval  30730  sgnsf  30731
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