Theorem negsdi 28060

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 27991	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) ∈ No )
2	1	negsidd 28052	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = 0s )
3		negsid 28051	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s )
4		negsid 28051	. . . . 5 ⊢ (𝐵 ∈ No → (𝐵 +s ( -us ‘𝐵)) = 0s )
5	3, 4	oveqan12d 7375	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s ( -us ‘𝐴)) +s (𝐵 +s ( -us ‘𝐵))) = ( 0s +s 0s ))
6		0no 27819	. . . . 5 ⊢ 0s ∈ No
7		addslid 27978	. . . . 5 ⊢ ( 0s ∈ No → ( 0s +s 0s ) = 0s )
8	6, 7	ax-mp 5	. . . 4 ⊢ ( 0s +s 0s ) = 0s
9	5, 8	eqtr2di 2791	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 0s = ((𝐴 +s ( -us ‘𝐴)) +s (𝐵 +s ( -us ‘𝐵))))
10		simpl 483	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
11	10	negscld 28047	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘𝐴) ∈ No )
12		simpr 485	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
13	12	negscld 28047	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘𝐵) ∈ No )
14	10, 11, 12, 13	adds4d 28019	. . 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 2778	. 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 28047	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) ∈ No )
17		negscl 28046	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
18		negscl 28046	. . . 4 ⊢ (𝐵 ∈ No → ( -us ‘𝐵) ∈ No )
19		addscl 27991	. . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ ( -us ‘𝐵) ∈ No ) → (( -us ‘𝐴) +s ( -us ‘𝐵)) ∈ No )
20	17, 18, 19	syl2an 602	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) +s ( -us ‘𝐵)) ∈ No )
21	16, 20, 1	addscan1d 28010	. 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 233	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) = (( -us ‘𝐴) +s ( -us ‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ‘cfv 6485  (class class class)co 7356   No csur 27621   0_s c0s 27815   +_s cadds 27969   -u_s cnegs 28029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-ot 4564  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-nadd 8592  df-no 27624  df-lts 27625  df-bday 27626  df-les 27727  df-slts 27768  df-cuts 27770  df-0s 27817  df-made 27837  df-old 27838  df-left 27840  df-right 27841  df-norec 27948  df-norec2 27959  df-adds 27970  df-negs 28031

This theorem is referenced by:  negsubsdi2d  28090  subsubs4d  28104  zcuts  28417  renegscl  28508  readdscl  28509