Theorem sgnsv 33791

Description: The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.)

Hypotheses

Ref	Expression
sgnsval.b	⊢ 𝐵 = (Base‘𝑅)
sgnsval.0	⊢ 0 = (0g‘𝑅)
sgnsval.l	⊢ < = (lt‘𝑅)
sgnsval.s	⊢ 𝑆 = (sgns‘𝑅)

Assertion

Ref	Expression
sgnsv	⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ 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.

Step	Hyp	Ref	Expression
1		sgnsval.s	. 2 ⊢ 𝑆 = (sgns‘𝑅)
2		elex 3475	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6908	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4		sgnsval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4	eqtr4di 2816	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6		fveq2 6908	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
7		sgnsval.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
8	6, 7	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
9	8	adantr 486	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (0g‘𝑟) = 0 )
10	9	eqeq2d 2774	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (𝑥 = (0g‘𝑟) ↔ 𝑥 = 0 ))
11		fveq2 6908	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = (lt‘𝑅))
12		sgnsval.l	. . . . . . . . . 10 ⊢ < = (lt‘𝑅)
13	11, 12	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = < )
14	13	adantr 486	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (lt‘𝑟) = < )
15		eqidd 2764	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → 𝑥 = 𝑥)
16	9, 14, 15	breq123d 5120	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → ((0g‘𝑟)(lt‘𝑟)𝑥 ↔ 0 < 𝑥))
17	16	ifbid 4509	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1) = if( 0 < 𝑥, 1, -1))
18	10, 17	ifbieq2d 4512	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1)) = if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))
19	5, 18	mpteq12dva 5194	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1))) = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
20		df-sgns 33790	. . . 4 ⊢ sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1))))
21	19, 20, 4	mptfvmpt 7257	. . 3 ⊢ (𝑅 ∈ V → (sgns‘𝑅) = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
22	2, 21	syl 18	. 2 ⊢ (𝑅 ∈ 𝑉 → (sgns‘𝑅) = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
23	1, 22	eqtrid 2810	1 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ifcif 4485   class class class wbr 5106   ↦ cmpt 5189  ‘cfv 6558  0cc0 11227  1c1 11228   -cneg 11569  Basecbs 17416  0_gc0g 17639  ltcplt 18511  sgn_scsgns 33789

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-sgns 33790

This theorem is used by:  sgnsval  33792  sgnsf  33793