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Theorem sgnsv 30061
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 3361 . . 3 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6332 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 sgnsval.b . . . . . 6 𝐵 = (Base‘𝑅)
53, 4syl6eqr 2822 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6 fveq2 6332 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
7 sgnsval.0 . . . . . . . . 9 0 = (0g𝑅)
86, 7syl6eqr 2822 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
98adantr 466 . . . . . . 7 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (0g𝑟) = 0 )
109eqeq2d 2780 . . . . . 6 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (𝑥 = (0g𝑟) ↔ 𝑥 = 0 ))
11 fveq2 6332 . . . . . . . . . 10 (𝑟 = 𝑅 → (lt‘𝑟) = (lt‘𝑅))
12 sgnsval.l . . . . . . . . . 10 < = (lt‘𝑅)
1311, 12syl6eqr 2822 . . . . . . . . 9 (𝑟 = 𝑅 → (lt‘𝑟) = < )
1413adantr 466 . . . . . . . 8 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → (lt‘𝑟) = < )
15 eqidd 2771 . . . . . . . 8 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → 𝑥 = 𝑥)
169, 14, 15breq123d 4798 . . . . . . 7 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → ((0g𝑟)(lt‘𝑟)𝑥0 < 𝑥))
1716ifbid 4245 . . . . . 6 ((𝑟 = 𝑅𝑥 ∈ (Base‘𝑟)) → if((0g𝑟)(lt‘𝑟)𝑥, 1, -1) = if( 0 < 𝑥, 1, -1))
1810, 17ifbieq2d 4248 . . . . 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 4864 . . . 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 30060 . . . 4 sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1))))
21 fvex 6342 . . . . 5 (Base‘𝑟) ∈ V
2221mptex 6629 . . . 4 (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1))) ∈ V
2319, 20, 22fvmpt3i 6429 . . 3 (𝑅 ∈ V → (sgns𝑅) = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
242, 23syl 17 . 2 (𝑅𝑉 → (sgns𝑅) = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
251, 24syl5eq 2816 1 (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  ifcif 4223   class class class wbr 4784  cmpt 4861  cfv 6031  0cc0 10137  1c1 10138  -cneg 10468  Basecbs 16063  0gc0g 16307  ltcplt 17148  sgnscsgns 30059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-sgns 30060
This theorem is referenced by:  sgnsval  30062  sgnsf  30063
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