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Theorem mulsgt0 28170
Description: The product of two positive surreals is positive. Theorem 9 of [Conway] p. 20. (Contributed by Scott Fenton, 6-Mar-2025.)
Assertion
Ref Expression
mulsgt0 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → 0s <s (𝐴 ·s 𝐵))

Proof of Theorem mulsgt0
StepHypRef Expression
1 0sno 27871 . . . 4 0s No
21a1i 11 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → 0s No )
3 simpll 767 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → 𝐴 No )
4 simprl 771 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → 𝐵 No )
5 simplr 769 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → 0s <s 𝐴)
6 simprr 773 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → 0s <s 𝐵)
72, 3, 2, 4, 5, 6sltmuld 28163 . 2 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) <s ((𝐴 ·s 𝐵) -s (𝐴 ·s 0s )))
8 muls02 28167 . . . . 5 (𝐵 No → ( 0s ·s 𝐵) = 0s )
94, 8syl 17 . . . 4 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → ( 0s ·s 𝐵) = 0s )
10 muls02 28167 . . . . . 6 ( 0s No → ( 0s ·s 0s ) = 0s )
111, 10ax-mp 5 . . . . 5 ( 0s ·s 0s ) = 0s
1211a1i 11 . . . 4 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → ( 0s ·s 0s ) = 0s )
139, 12oveq12d 7449 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = ( 0s -s 0s ))
14 subsid 28099 . . . 4 ( 0s No → ( 0s -s 0s ) = 0s )
151, 14ax-mp 5 . . 3 ( 0s -s 0s ) = 0s
1613, 15eqtrdi 2793 . 2 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = 0s )
17 muls01 28138 . . . . 5 (𝐴 No → (𝐴 ·s 0s ) = 0s )
183, 17syl 17 . . . 4 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → (𝐴 ·s 0s ) = 0s )
1918oveq2d 7447 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → ((𝐴 ·s 𝐵) -s (𝐴 ·s 0s )) = ((𝐴 ·s 𝐵) -s 0s ))
20 mulscl 28160 . . . . 5 ((𝐴 No 𝐵 No ) → (𝐴 ·s 𝐵) ∈ No )
2120ad2ant2r 747 . . . 4 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → (𝐴 ·s 𝐵) ∈ No )
22 subsid1 28098 . . . 4 ((𝐴 ·s 𝐵) ∈ No → ((𝐴 ·s 𝐵) -s 0s ) = (𝐴 ·s 𝐵))
2321, 22syl 17 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → ((𝐴 ·s 𝐵) -s 0s ) = (𝐴 ·s 𝐵))
2419, 23eqtrd 2777 . 2 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → ((𝐴 ·s 𝐵) -s (𝐴 ·s 0s )) = (𝐴 ·s 𝐵))
257, 16, 243brtr3d 5174 1 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → 0s <s (𝐴 ·s 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108   class class class wbr 5143  (class class class)co 7431   No csur 27684   <s cslt 27685   0s c0s 27867   -s csubs 28052   ·s cmuls 28132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133
This theorem is referenced by:  mulsgt0d  28171
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