Theorem negsdi 28143

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 28074	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) ∈ No )
2	1	negsidd 28135	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = 0s )
3		negsid 28134	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s )
4		negsid 28134	. . . . 5 ⊢ (𝐵 ∈ No → (𝐵 +s ( -us ‘𝐵)) = 0s )
5	3, 4	oveqan12d 7415	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s ( -us ‘𝐴)) +s (𝐵 +s ( -us ‘𝐵))) = ( 0s +s 0s ))
6		0no 27902	. . . . 5 ⊢ 0s ∈ No
7		addslid 28061	. . . . 5 ⊢ ( 0s ∈ No → ( 0s +s 0s ) = 0s )
8	6, 7	ax-mp 5	. . . 4 ⊢ ( 0s +s 0s ) = 0s
9	5, 8	eqtr2di 2814	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 0s = ((𝐴 +s ( -us ‘𝐴)) +s (𝐵 +s ( -us ‘𝐵))))
10		simpl 486	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
11	10	negscld 28130	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘𝐴) ∈ No )
12		simpr 488	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
13	12	negscld 28130	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘𝐵) ∈ No )
14	10, 11, 12, 13	adds4d 28102	. . 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 2801	. 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 28130	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) ∈ No )
17		negscl 28129	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
18		negscl 28129	. . . 4 ⊢ (𝐵 ∈ No → ( -us ‘𝐵) ∈ No )
19		addscl 28074	. . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ ( -us ‘𝐵) ∈ No ) → (( -us ‘𝐴) +s ( -us ‘𝐵)) ∈ No )
20	17, 18, 19	syl2an 605	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) +s ( -us ‘𝐵)) ∈ No )
21	16, 20, 1	addscan1d 28093	. 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 234	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) = (( -us ‘𝐴) +s ( -us ‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ‘cfv 6521  (class class class)co 7396   No csur 27704   0_s c0s 27898   +_s cadds 28052   -u_s cnegs 28112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-1o 8437  df-2o 8438  df-nadd 8636  df-no 27707  df-lts 27708  df-bday 27709  df-les 27809  df-slts 27851  df-cuts 27853  df-0s 27900  df-made 27920  df-old 27921  df-left 27923  df-right 27924  df-norec 28031  df-norec2 28042  df-adds 28053  df-negs 28114

This theorem is referenced by:  negsubsdi2d  28173  subsubs4d  28187  zcuts  28500  renegscl  28591  readdscl  28592