Theorem negsdi 28058

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 27989	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) ∈ No )
2	1	negsidd 28050	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = 0s )
3		negsid 28049	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 +s ( -us ‘𝐴)) = 0s )
4		negsid 28049	. . . . 5 ⊢ (𝐵 ∈ No → (𝐵 +s ( -us ‘𝐵)) = 0s )
5	3, 4	oveqan12d 7387	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s ( -us ‘𝐴)) +s (𝐵 +s ( -us ‘𝐵))) = ( 0s +s 0s ))
6		0no 27817	. . . . 5 ⊢ 0s ∈ No
7		addslid 27976	. . . . 5 ⊢ ( 0s ∈ No → ( 0s +s 0s ) = 0s )
8	6, 7	ax-mp 5	. . . 4 ⊢ ( 0s +s 0s ) = 0s
9	5, 8	eqtr2di 2789	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 0s = ((𝐴 +s ( -us ‘𝐴)) +s (𝐵 +s ( -us ‘𝐵))))
10		simpl 482	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
11	10	negscld 28045	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘𝐴) ∈ No )
12		simpr 484	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
13	12	negscld 28045	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘𝐵) ∈ No )
14	10, 11, 12, 13	adds4d 28017	. . 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 2776	. 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 28045	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) ∈ No )
17		negscl 28044	. . . 4 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
18		negscl 28044	. . . 4 ⊢ (𝐵 ∈ No → ( -us ‘𝐵) ∈ No )
19		addscl 27989	. . . 4 ⊢ ((( -us ‘𝐴) ∈ No ∧ ( -us ‘𝐵) ∈ No ) → (( -us ‘𝐴) +s ( -us ‘𝐵)) ∈ No )
20	17, 18, 19	syl2an 597	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) +s ( -us ‘𝐵)) ∈ No )
21	16, 20, 1	addscan1d 28008	. 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 232	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) = (( -us ‘𝐴) +s ( -us ‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ‘cfv 6500  (class class class)co 7368   No csur 27619   0_s c0s 27813   +_s cadds 27967   -u_s cnegs 28027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-1o 8407  df-2o 8408  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029

This theorem is referenced by:  negsubsdi2d  28088  subsubs4d  28102  zcuts  28415  renegscl  28506  readdscl  28507