Theorem zaddscl 28545

Description: The surreal integers are closed under addition. (Contributed by Scott Fenton, 25-Jul-2025.)

Assertion

Ref	Expression
zaddscl	⊢ ((𝐴 ∈ ℤs ∧ 𝐵 ∈ ℤs) → (𝐴 +s 𝐵) ∈ ℤs)

Proof of Theorem zaddscl

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reeanv 3237	. . 3 ⊢ (∃𝑥 ∈ ℕs ∃𝑧 ∈ ℕs (∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)) ↔ (∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑧 ∈ ℕs ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)))
2		reeanv 3237	. . . 4 ⊢ (∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) ↔ (∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)))
3	2	2rexbii 3141	. . 3 ⊢ (∃𝑥 ∈ ℕs ∃𝑧 ∈ ℕs ∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) ↔ ∃𝑥 ∈ ℕs ∃𝑧 ∈ ℕs (∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)))
4		elzs 28535	. . . 4 ⊢ (𝐴 ∈ ℤs ↔ ∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦))
5		elzs 28535	. . . 4 ⊢ (𝐵 ∈ ℤs ↔ ∃𝑧 ∈ ℕs ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤))
6	4, 5	anbi12i 639	. . 3 ⊢ ((𝐴 ∈ ℤs ∧ 𝐵 ∈ ℤs) ↔ (∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑧 ∈ ℕs ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)))
7	1, 3, 6	3bitr4ri 307	. 2 ⊢ ((𝐴 ∈ ℤs ∧ 𝐵 ∈ ℤs) ↔ ∃𝑥 ∈ ℕs ∃𝑧 ∈ ℕs ∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)))
8		simpll 778	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑥 ∈ ℕs)
9	8	nnnod 28477	. . . . . . 7 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑥 ∈ No )
10		simplr 780	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑧 ∈ ℕs)
11	10	nnnod 28477	. . . . . . 7 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑧 ∈ No )
12		simprl 782	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑦 ∈ ℕs)
13	12	nnnod 28477	. . . . . . 7 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑦 ∈ No )
14		simprr 784	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑤 ∈ ℕs)
15	14	nnnod 28477	. . . . . . 7 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑤 ∈ No )
16	9, 11, 13, 15	addsubs4d 28252	. . . . . 6 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 +s 𝑧) -s (𝑦 +s 𝑤)) = ((𝑥 -s 𝑦) +s (𝑧 -s 𝑤)))
17		nnaddscl 28497	. . . . . . 7 ⊢ ((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) → (𝑥 +s 𝑧) ∈ ℕs)
18		nnaddscl 28497	. . . . . . 7 ⊢ ((𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs) → (𝑦 +s 𝑤) ∈ ℕs)
19		nnzsubs 28536	. . . . . . 7 ⊢ (((𝑥 +s 𝑧) ∈ ℕs ∧ (𝑦 +s 𝑤) ∈ ℕs) → ((𝑥 +s 𝑧) -s (𝑦 +s 𝑤)) ∈ ℤs)
20	17, 18, 19	syl2an 607	. . . . . 6 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 +s 𝑧) -s (𝑦 +s 𝑤)) ∈ ℤs)
21	16, 20	eqeltrrd 2866	. . . . 5 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) +s (𝑧 -s 𝑤)) ∈ ℤs)
22		oveq12 7409	. . . . . 6 ⊢ ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 +s 𝐵) = ((𝑥 -s 𝑦) +s (𝑧 -s 𝑤)))
23	22	eleq1d 2850	. . . . 5 ⊢ ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → ((𝐴 +s 𝐵) ∈ ℤs ↔ ((𝑥 -s 𝑦) +s (𝑧 -s 𝑤)) ∈ ℤs))
24	21, 23	syl5ibrcom 250	. . . 4 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 +s 𝐵) ∈ ℤs))
25	24	rexlimdvva 3222	. . 3 ⊢ ((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) → (∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 +s 𝐵) ∈ ℤs))
26	25	rexlimivv 3207	. 2 ⊢ (∃𝑥 ∈ ℕs ∃𝑧 ∈ ℕs ∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 +s 𝐵) ∈ ℤs)
27	7, 26	sylbi 220	1 ⊢ ((𝐴 ∈ ℤs ∧ 𝐵 ∈ ℤs) → (𝐴 +s 𝐵) ∈ ℤs)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∃wrex 3089  (class class class)co 7400   +_s cadds 28110   -_s csubs 28171  ℕ_scnns 28464  ℤ_sczs 28529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-nadd 8640  df-no 27765  df-lts 27766  df-bday 27767  df-les 27867  df-slts 27909  df-cuts 27911  df-0s 27958  df-1s 27959  df-made 27978  df-old 27979  df-left 27981  df-right 27982  df-norec 28089  df-norec2 28100  df-adds 28111  df-negs 28172  df-subs 28173  df-n0s 28465  df-nns 28466  df-zs 28530

This theorem is referenced by:  zaddscld  28546  zsoring  28560  pw2cutp1  28612