Theorem negsdi 28296

Description: Distribution of surreal negative over addition. (Contributed by Scott Fenton, 5-Feb-2025.)

Assertion

Ref	Expression
negsdi	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) = (( -us ‘𝐴) +s ( -us ‘𝐵)))

Proof of Theorem negsdi

Step	Hyp	Ref	Expression
1		addscl 28227	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) ∈ No )
2	1	negsidd 28288	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = 0s )
3		negsid 28287	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s )
4		negsid 28287	. . . . 5 ⊢ (𝐵 ∈ No → (𝐵 +s ( -us ‘𝐵)) = 0s )
5	3, 4	oveqan12d 7438	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s ( -us ‘𝐴)) +s (𝐵 +s ( -us ‘𝐵))) = ( 0s +s 0s ))
6		0no 28055	. . . . 5 ⊢ 0s ∈ No
7		addslid 28214	. . . . 5 ⊢ ( 0s ∈ No → ( 0s +s 0s ) = 0s )
8	6, 7	ax-mp 5	. . . 4 ⊢ ( 0s +s 0s ) = 0s
9	5, 8	eqtr2di 2817	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 0s = ((𝐴 +s ( -us ‘𝐴)) +s (𝐵 +s ( -us ‘𝐵))))
10		simpl 488	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
11	10	negscld 28283	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘𝐴) ∈ No )
12		simpr 490	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
13	12	negscld 28283	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘𝐵) ∈ No )
14	10, 11, 12, 13	adds4d 28255	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s ( -us ‘𝐴)) +s (𝐵 +s ( -us ‘𝐵))) = ((𝐴 +s 𝐵) +s (( -us ‘𝐴) +s ( -us ‘𝐵))))
15	2, 9, 14	3eqtrd 2804	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = ((𝐴 +s 𝐵) +s (( -us ‘𝐴) +s ( -us ‘𝐵))))
16	1	negscld 28283	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) ∈ No )
17		negscl 28282	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
18		negscl 28282	. . . 4 ⊢ (𝐵 ∈ No → ( -us ‘𝐵) ∈ No )
19		addscl 28227	. . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ ( -us ‘𝐵) ∈ No ) → (( -us ‘𝐴) +s ( -us ‘𝐵)) ∈ No )
20	17, 18, 19	syl2an 608	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) +s ( -us ‘𝐵)) ∈ No )
21	16, 20, 1	addscan1d 28246	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = ((𝐴 +s 𝐵) +s (( -us ‘𝐴) +s ( -us ‘𝐵))) ↔ ( -us ‘(𝐴 +s 𝐵)) = (( -us ‘𝐴) +s ( -us ‘𝐵))))
22	15, 21	mpbid 235	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) = (( -us ‘𝐴) +s ( -us ‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ‘cfv 6540  (class class class)co 7419   No csur 27857   0_s c0s 28051   +_s cadds 28205   -u_s cnegs 28265

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-1o 8459  df-2o 8460  df-nadd 8658  df-no 27860  df-lts 27861  df-bday 27862  df-les 27962  df-slts 28004  df-cuts 28006  df-0s 28053  df-made 28073  df-old 28074  df-left 28076  df-right 28077  df-norec 28184  df-norec2 28195  df-adds 28206  df-negs 28267

This theorem is used by:  negsubsdi2d  28326  subsubs4d  28340  zcuts  28653  renegscl  28744  readdscl  28745