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Theorem n0mulscl 28349
Description: The non-negative surreal integers are closed under multiplication. (Contributed by Scott Fenton, 15-Apr-2025.)
Assertion
Ref Expression
n0mulscl ((𝐴 ∈ ℕ0s𝐵 ∈ ℕ0s) → (𝐴 ·s 𝐵) ∈ ℕ0s)

Proof of Theorem n0mulscl
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7440 . . . . 5 (𝑛 = 0s → (𝐴 ·s 𝑛) = (𝐴 ·s 0s ))
21eleq1d 2825 . . . 4 (𝑛 = 0s → ((𝐴 ·s 𝑛) ∈ ℕ0s ↔ (𝐴 ·s 0s ) ∈ ℕ0s))
32imbi2d 340 . . 3 (𝑛 = 0s → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) ∈ ℕ0s)))
4 oveq2 7440 . . . . 5 (𝑛 = 𝑚 → (𝐴 ·s 𝑛) = (𝐴 ·s 𝑚))
54eleq1d 2825 . . . 4 (𝑛 = 𝑚 → ((𝐴 ·s 𝑛) ∈ ℕ0s ↔ (𝐴 ·s 𝑚) ∈ ℕ0s))
65imbi2d 340 . . 3 (𝑛 = 𝑚 → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 ·s 𝑚) ∈ ℕ0s)))
7 oveq2 7440 . . . . 5 (𝑛 = (𝑚 +s 1s ) → (𝐴 ·s 𝑛) = (𝐴 ·s (𝑚 +s 1s )))
87eleq1d 2825 . . . 4 (𝑛 = (𝑚 +s 1s ) → ((𝐴 ·s 𝑛) ∈ ℕ0s ↔ (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s))
98imbi2d 340 . . 3 (𝑛 = (𝑚 +s 1s ) → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s)))
10 oveq2 7440 . . . . 5 (𝑛 = 𝐵 → (𝐴 ·s 𝑛) = (𝐴 ·s 𝐵))
1110eleq1d 2825 . . . 4 (𝑛 = 𝐵 → ((𝐴 ·s 𝑛) ∈ ℕ0s ↔ (𝐴 ·s 𝐵) ∈ ℕ0s))
1211imbi2d 340 . . 3 (𝑛 = 𝐵 → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑛) ∈ ℕ0s) ↔ (𝐴 ∈ ℕ0s → (𝐴 ·s 𝐵) ∈ ℕ0s)))
13 n0sno 28329 . . . . 5 (𝐴 ∈ ℕ0s𝐴 No )
14 muls01 28139 . . . . 5 (𝐴 No → (𝐴 ·s 0s ) = 0s )
1513, 14syl 17 . . . 4 (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) = 0s )
16 0n0s 28335 . . . 4 0s ∈ ℕ0s
1715, 16eqeltrdi 2848 . . 3 (𝐴 ∈ ℕ0s → (𝐴 ·s 0s ) ∈ ℕ0s)
1813ad2antrr 726 . . . . . . . . 9 (((𝐴 ∈ ℕ0s𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 𝐴 No )
19 n0sno 28329 . . . . . . . . . 10 (𝑚 ∈ ℕ0s𝑚 No )
2019ad2antlr 727 . . . . . . . . 9 (((𝐴 ∈ ℕ0s𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 𝑚 No )
21 1sno 27873 . . . . . . . . . 10 1s No
2221a1i 11 . . . . . . . . 9 (((𝐴 ∈ ℕ0s𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → 1s No )
2318, 20, 22addsdid 28183 . . . . . . . 8 (((𝐴 ∈ ℕ0s𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ·s (𝑚 +s 1s )) = ((𝐴 ·s 𝑚) +s (𝐴 ·s 1s )))
2413mulsridd 28141 . . . . . . . . . 10 (𝐴 ∈ ℕ0s → (𝐴 ·s 1s ) = 𝐴)
2524oveq2d 7448 . . . . . . . . 9 (𝐴 ∈ ℕ0s → ((𝐴 ·s 𝑚) +s (𝐴 ·s 1s )) = ((𝐴 ·s 𝑚) +s 𝐴))
2625ad2antrr 726 . . . . . . . 8 (((𝐴 ∈ ℕ0s𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → ((𝐴 ·s 𝑚) +s (𝐴 ·s 1s )) = ((𝐴 ·s 𝑚) +s 𝐴))
2723, 26eqtrd 2776 . . . . . . 7 (((𝐴 ∈ ℕ0s𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ·s (𝑚 +s 1s )) = ((𝐴 ·s 𝑚) +s 𝐴))
28 n0addscl 28348 . . . . . . . . 9 (((𝐴 ·s 𝑚) ∈ ℕ0s𝐴 ∈ ℕ0s) → ((𝐴 ·s 𝑚) +s 𝐴) ∈ ℕ0s)
2928ancoms 458 . . . . . . . 8 ((𝐴 ∈ ℕ0s ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → ((𝐴 ·s 𝑚) +s 𝐴) ∈ ℕ0s)
3029adantlr 715 . . . . . . 7 (((𝐴 ∈ ℕ0s𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → ((𝐴 ·s 𝑚) +s 𝐴) ∈ ℕ0s)
3127, 30eqeltrd 2840 . . . . . 6 (((𝐴 ∈ ℕ0s𝑚 ∈ ℕ0s) ∧ (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s)
3231ex 412 . . . . 5 ((𝐴 ∈ ℕ0s𝑚 ∈ ℕ0s) → ((𝐴 ·s 𝑚) ∈ ℕ0s → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s))
3332expcom 413 . . . 4 (𝑚 ∈ ℕ0s → (𝐴 ∈ ℕ0s → ((𝐴 ·s 𝑚) ∈ ℕ0s → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s)))
3433a2d 29 . . 3 (𝑚 ∈ ℕ0s → ((𝐴 ∈ ℕ0s → (𝐴 ·s 𝑚) ∈ ℕ0s) → (𝐴 ∈ ℕ0s → (𝐴 ·s (𝑚 +s 1s )) ∈ ℕ0s)))
353, 6, 9, 12, 17, 34n0sind 28338 . 2 (𝐵 ∈ ℕ0s → (𝐴 ∈ ℕ0s → (𝐴 ·s 𝐵) ∈ ℕ0s))
3635impcom 407 1 ((𝐴 ∈ ℕ0s𝐵 ∈ ℕ0s) → (𝐴 ·s 𝐵) ∈ ℕ0s)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  (class class class)co 7432   No csur 27685   0s c0s 27868   1s c1s 27869   +s cadds 27993   ·s cmuls 28133  0scnn0s 28319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-1s 27871  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-norec2 27983  df-adds 27994  df-negs 28054  df-subs 28055  df-muls 28134  df-n0s 28321
This theorem is referenced by:  nnmulscl  28351  addhalfcut  28420
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