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Theorem mulsgt0 28185
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 27886 . . . 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 28178 . 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 28182 . . . . 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 28182 . . . . . 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 28114 . . . 4 ( 0s No → ( 0s -s 0s ) = 0s )
151, 14ax-mp 5 . . 3 ( 0s -s 0s ) = 0s
1613, 15eqtrdi 2791 . 2 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = 0s )
17 muls01 28153 . . . . 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 28175 . . . . 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 28113 . . . 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 2775 . 2 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → ((𝐴 ·s 𝐵) -s (𝐴 ·s 0s )) = (𝐴 ·s 𝐵))
257, 16, 243brtr3d 5179 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 1537  wcel 2106   class class class wbr 5148  (class class class)co 7431   No csur 27699   <s cslt 27700   0s c0s 27882   -s csubs 28067   ·s cmuls 28147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148
This theorem is referenced by:  mulsgt0d  28186
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