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Theorem zmulscld 28403
Description: The surreal integers are closed under multiplication. (Contributed by Scott Fenton, 20-Aug-2025.)
Hypotheses
Ref Expression
zmulscld.1 (𝜑𝐴 ∈ ℤs)
zmulscld.2 (𝜑𝐵 ∈ ℤs)
Assertion
Ref Expression
zmulscld (𝜑 → (𝐴 ·s 𝐵) ∈ ℤs)

Proof of Theorem zmulscld
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zmulscld.1 . . 3 (𝜑𝐴 ∈ ℤs)
2 elzs 28390 . . 3 (𝐴 ∈ ℤs ↔ ∃𝑥 ∈ ℕs𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦))
31, 2sylib 218 . 2 (𝜑 → ∃𝑥 ∈ ℕs𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦))
4 zmulscld.2 . . 3 (𝜑𝐵 ∈ ℤs)
5 elzs 28390 . . 3 (𝐵 ∈ ℤs ↔ ∃𝑧 ∈ ℕs𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤))
64, 5sylib 218 . 2 (𝜑 → ∃𝑧 ∈ ℕs𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤))
7 reeanv 3235 . . . . 5 (∃𝑦 ∈ ℕs𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) ↔ (∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)))
872rexbii 3135 . . . 4 (∃𝑥 ∈ ℕs𝑧 ∈ ℕs𝑦 ∈ ℕs𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) ↔ ∃𝑥 ∈ ℕs𝑧 ∈ ℕs (∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)))
9 reeanv 3235 . . . 4 (∃𝑥 ∈ ℕs𝑧 ∈ ℕs (∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)) ↔ (∃𝑥 ∈ ℕs𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑧 ∈ ℕs𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)))
108, 9bitri 275 . . 3 (∃𝑥 ∈ ℕs𝑧 ∈ ℕs𝑦 ∈ ℕs𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) ↔ (∃𝑥 ∈ ℕs𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑧 ∈ ℕs𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)))
11 nnsno 28349 . . . . . . . . . . 11 (𝑥 ∈ ℕs𝑥 No )
1211ad2antrr 725 . . . . . . . . . 10 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → 𝑥 No )
13 nnsno 28349 . . . . . . . . . . 11 (𝑦 ∈ ℕs𝑦 No )
1413ad2antrl 727 . . . . . . . . . 10 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → 𝑦 No )
1512, 14subscld 28113 . . . . . . . . 9 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (𝑥 -s 𝑦) ∈ No )
16 nnsno 28349 . . . . . . . . . 10 (𝑧 ∈ ℕs𝑧 No )
1716ad2antlr 726 . . . . . . . . 9 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → 𝑧 No )
18 nnsno 28349 . . . . . . . . . 10 (𝑤 ∈ ℕs𝑤 No )
1918ad2antll 728 . . . . . . . . 9 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → 𝑤 No )
2015, 17, 19subsdid 28204 . . . . . . . 8 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)) = (((𝑥 -s 𝑦) ·s 𝑧) -s ((𝑥 -s 𝑦) ·s 𝑤)))
21 nnmulscl 28370 . . . . . . . . . . . . 13 ((𝑥 ∈ ℕs𝑧 ∈ ℕs) → (𝑥 ·s 𝑧) ∈ ℕs)
2221adantr 480 . . . . . . . . . . . 12 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (𝑥 ·s 𝑧) ∈ ℕs)
2322nnsnod 28351 . . . . . . . . . . 11 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (𝑥 ·s 𝑧) ∈ No )
24 simprl 770 . . . . . . . . . . . . 13 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → 𝑦 ∈ ℕs)
25 simplr 768 . . . . . . . . . . . . 13 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → 𝑧 ∈ ℕs)
26 nnmulscl 28370 . . . . . . . . . . . . 13 ((𝑦 ∈ ℕs𝑧 ∈ ℕs) → (𝑦 ·s 𝑧) ∈ ℕs)
2724, 25, 26syl2anc 583 . . . . . . . . . . . 12 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (𝑦 ·s 𝑧) ∈ ℕs)
2827nnsnod 28351 . . . . . . . . . . 11 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (𝑦 ·s 𝑧) ∈ No )
2923, 28subscld 28113 . . . . . . . . . 10 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → ((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)) ∈ No )
30 nnmulscl 28370 . . . . . . . . . . . 12 ((𝑥 ∈ ℕs𝑤 ∈ ℕs) → (𝑥 ·s 𝑤) ∈ ℕs)
3130ad2ant2rl 748 . . . . . . . . . . 11 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (𝑥 ·s 𝑤) ∈ ℕs)
3231nnsnod 28351 . . . . . . . . . 10 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (𝑥 ·s 𝑤) ∈ No )
33 nnmulscl 28370 . . . . . . . . . . . 12 ((𝑦 ∈ ℕs𝑤 ∈ ℕs) → (𝑦 ·s 𝑤) ∈ ℕs)
3433adantl 481 . . . . . . . . . . 11 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (𝑦 ·s 𝑤) ∈ ℕs)
3534nnsnod 28351 . . . . . . . . . 10 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (𝑦 ·s 𝑤) ∈ No )
3629, 32, 35subsubs2d 28145 . . . . . . . . 9 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)) -s ((𝑥 ·s 𝑤) -s (𝑦 ·s 𝑤))) = (((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)) +s ((𝑦 ·s 𝑤) -s (𝑥 ·s 𝑤))))
3712, 14, 17subsdird 28205 . . . . . . . . . 10 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) ·s 𝑧) = ((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)))
3812, 14, 19subsdird 28205 . . . . . . . . . 10 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) ·s 𝑤) = ((𝑥 ·s 𝑤) -s (𝑦 ·s 𝑤)))
3937, 38oveq12d 7468 . . . . . . . . 9 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (((𝑥 -s 𝑦) ·s 𝑧) -s ((𝑥 -s 𝑦) ·s 𝑤)) = (((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)) -s ((𝑥 ·s 𝑤) -s (𝑦 ·s 𝑤))))
4023, 35, 28, 32addsubs4d 28150 . . . . . . . . 9 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (((𝑥 ·s 𝑧) -s (𝑦 ·s 𝑧)) +s ((𝑦 ·s 𝑤) -s (𝑥 ·s 𝑤))))
4136, 39, 403eqtr4d 2790 . . . . . . . 8 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (((𝑥 -s 𝑦) ·s 𝑧) -s ((𝑥 -s 𝑦) ·s 𝑤)) = (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))))
4220, 41eqtrd 2780 . . . . . . 7 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)) = (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))))
43 nnaddscl 28369 . . . . . . . . . 10 (((𝑥 ·s 𝑧) ∈ ℕs ∧ (𝑦 ·s 𝑤) ∈ ℕs) → ((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) ∈ ℕs)
4422, 34, 43syl2anc 583 . . . . . . . . 9 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → ((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) ∈ ℕs)
45 nnaddscl 28369 . . . . . . . . . 10 (((𝑦 ·s 𝑧) ∈ ℕs ∧ (𝑥 ·s 𝑤) ∈ ℕs) → ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤)) ∈ ℕs)
4627, 31, 45syl2anc 583 . . . . . . . . 9 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤)) ∈ ℕs)
47 eqid 2740 . . . . . . . . . 10 (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤)))
48 rspceov 7499 . . . . . . . . . 10 ((((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) ∈ ℕs ∧ ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤)) ∈ ℕs ∧ (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤)))) → ∃𝑡 ∈ ℕs𝑢 ∈ ℕs (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (𝑡 -s 𝑢))
4947, 48mp3an3 1450 . . . . . . . . 9 ((((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) ∈ ℕs ∧ ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤)) ∈ ℕs) → ∃𝑡 ∈ ℕs𝑢 ∈ ℕs (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (𝑡 -s 𝑢))
5044, 46, 49syl2anc 583 . . . . . . . 8 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → ∃𝑡 ∈ ℕs𝑢 ∈ ℕs (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (𝑡 -s 𝑢))
51 elzs 28390 . . . . . . . 8 ((((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) ∈ ℤs ↔ ∃𝑡 ∈ ℕs𝑢 ∈ ℕs (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) = (𝑡 -s 𝑢))
5250, 51sylibr 234 . . . . . . 7 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → (((𝑥 ·s 𝑧) +s (𝑦 ·s 𝑤)) -s ((𝑦 ·s 𝑧) +s (𝑥 ·s 𝑤))) ∈ ℤs)
5342, 52eqeltrd 2844 . . . . . 6 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)) ∈ ℤs)
54 oveq12 7459 . . . . . . 7 ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) = ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)))
5554eleq1d 2829 . . . . . 6 ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → ((𝐴 ·s 𝐵) ∈ ℤs ↔ ((𝑥 -s 𝑦) ·s (𝑧 -s 𝑤)) ∈ ℤs))
5653, 55syl5ibrcom 247 . . . . 5 (((𝑥 ∈ ℕs𝑧 ∈ ℕs) ∧ (𝑦 ∈ ℕs𝑤 ∈ ℕs)) → ((𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) ∈ ℤs))
5756rexlimdvva 3219 . . . 4 ((𝑥 ∈ ℕs𝑧 ∈ ℕs) → (∃𝑦 ∈ ℕs𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) ∈ ℤs))
5857rexlimivv 3207 . . 3 (∃𝑥 ∈ ℕs𝑧 ∈ ℕs𝑦 ∈ ℕs𝑤 ∈ ℕs (𝐴 = (𝑥 -s 𝑦) ∧ 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) ∈ ℤs)
5910, 58sylbir 235 . 2 ((∃𝑥 ∈ ℕs𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦) ∧ ∃𝑧 ∈ ℕs𝑤 ∈ ℕs 𝐵 = (𝑧 -s 𝑤)) → (𝐴 ·s 𝐵) ∈ ℤs)
603, 6, 59syl2anc 583 1 (𝜑 → (𝐴 ·s 𝐵) ∈ ℤs)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  wrex 3076  (class class class)co 7450   No csur 27704   +s cadds 28012   -s csubs 28072   ·s cmuls 28152  scnns 28339  sczs 28384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-riota 7406  df-ov 7453  df-oprab 7454  df-mpo 7455  df-om 7906  df-1st 8032  df-2nd 8033  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-1o 8524  df-2o 8525  df-nadd 8724  df-no 27707  df-slt 27708  df-bday 27709  df-sle 27810  df-sslt 27846  df-scut 27848  df-0s 27889  df-1s 27890  df-made 27906  df-old 27907  df-left 27909  df-right 27910  df-norec 27991  df-norec2 28002  df-adds 28013  df-negs 28073  df-subs 28074  df-muls 28153  df-n0s 28340  df-nns 28341  df-zs 28385
This theorem is referenced by:  zs12bday  28444
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