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 = (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 3482	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6901	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4		sgnsval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4	eqtr4di 2784	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6		fveq2 6901	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
7		sgnsval.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
8	6, 7	eqtr4di 2784	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
9	8	adantr 479	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (0g‘𝑟) = 0 )
10	9	eqeq2d 2737	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (𝑥 = (0g‘𝑟) ↔ 𝑥 = 0 ))
11		fveq2 6901	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = (lt‘𝑅))
12		sgnsval.l	. . . . . . . . . 10 ⊢ < = (lt‘𝑅)
13	11, 12	eqtr4di 2784	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (lt‘𝑟) = < )
14	13	adantr 479	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → (lt‘𝑟) = < )
15		eqidd 2727	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → 𝑥 = 𝑥)
16	9, 14, 15	breq123d 5167	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → ((0g‘𝑟)(lt‘𝑟)𝑥 ↔ 0 < 𝑥))
17	16	ifbid 4556	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑥 ∈ (Base‘𝑟)) → if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1) = if( 0 < 𝑥, 1, -1))
18	10, 17	ifbieq2d 4559	. . . . 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 5242	. . . 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 33037	. . . 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 7245	. . 3 ⊢ (𝑅 ∈ V → (sgns‘𝑅) = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
22	2, 21	syl 17	. 2 ⊢ (𝑅 ∈ 𝑉 → (sgns‘𝑅) = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
23	1, 22	eqtrid 2778	1 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))

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  0_gc0g 17454  ltcplt 18333  sgn_scsgns 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