Theorem zaddscl 28457

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 3228	. . 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 3228	. . . 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 3132	. . 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 28447	. . . 4 ⊢ (𝐴 ∈ ℤs ↔ ∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦))
5		elzs 28447	. . . 4 ⊢ (𝐵 ∈ ℤs ↔ ∃𝑧 ∈ ℕs ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤))
6	4, 5	anbi12i 636	. . 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 306	. 2 ⊢ ((𝐴 ∈ ℤs ∧ 𝐵 ∈ ℤs) ↔ ∃𝑥 ∈ ℕs ∃𝑧 ∈ ℕs ∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)))
8		simpll 774	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑥 ∈ ℕs)
9	8	nnnod 28389	. . . . . . 7 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑥 ∈ No )
10		simplr 776	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑧 ∈ ℕs)
11	10	nnnod 28389	. . . . . . 7 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑧 ∈ No )
12		simprl 778	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑦 ∈ ℕs)
13	12	nnnod 28389	. . . . . . 7 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑦 ∈ No )
14		simprr 780	. . . . . . . 8 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑤 ∈ ℕs)
15	14	nnnod 28389	. . . . . . 7 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → 𝑤 ∈ No )
16	9, 11, 13, 15	addsubs4d 28164	. . . . . 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 28409	. . . . . . 7 ⊢ ((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) → (𝑥 +s 𝑧) ∈ ℕs)
18		nnaddscl 28409	. . . . . . 7 ⊢ ((𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs) → (𝑦 +s 𝑤) ∈ ℕs)
19		nnzsubs 28448	. . . . . . 7 ⊢ (((𝑥 +s 𝑧) ∈ ℕs ∧ (𝑦 +s 𝑤) ∈ ℕs) → ((𝑥 +s 𝑧) -s (𝑦 +s 𝑤)) ∈ ℤs)
20	17, 18, 19	syl2an 604	. . . . . 6 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 +s 𝑧) -s (𝑦 +s 𝑤)) ∈ ℤs)
21	16, 20	eqeltrrd 2857	. . . . 5 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) +s (𝑧 -s 𝑤)) ∈ ℤs)
22		oveq12 7394	. . . . . 6 ⊢ ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 +s 𝐵) = ((𝑥 -s 𝑦) +s (𝑧 -s 𝑤)))
23	22	eleq1d 2841	. . . . 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 249	. . . 4 ⊢ (((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs ∧ 𝑤 ∈ ℕs)) → ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 +s 𝐵) ∈ ℤs))
25	24	rexlimdvva 3213	. . 3 ⊢ ((𝑥 ∈ ℕs ∧ 𝑧 ∈ ℕs) → (∃𝑦 ∈ ℕs ∃𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 +s 𝐵) ∈ ℤs))
26	25	rexlimivv 3198	. 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 219	1 ⊢ ((𝐴 ∈ ℤs ∧ 𝐵 ∈ ℤs) → (𝐴 +s 𝐵) ∈ ℤs)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1554   ∈ wcel 2136  ∃wrex 3080  (class class class)co 7385   +_s cadds 28022   -_s csubs 28083  ℕ_scnns 28376  ℤ_sczs 28441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-ot 4585  df-uni 4860  df-int 4900  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-riota 7342  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-1o 8425  df-2o 8426  df-nadd 8624  df-no 27677  df-lts 27678  df-bday 27679  df-les 27779  df-slts 27821  df-cuts 27823  df-0s 27870  df-1s 27871  df-made 27890  df-old 27891  df-left 27893  df-right 27894  df-norec 28001  df-norec2 28012  df-adds 28023  df-negs 28084  df-subs 28085  df-n0s 28377  df-nns 28378  df-zs 28442

This theorem is referenced by:  zaddscld  28458  zsoring  28472  pw2cutp1  28524