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Theorem mulsgt0 27529
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 27256 . . . 4 0s No
21a1i 11 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → 0s No )
3 simpll 765 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → 𝐴 No )
4 simprl 769 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → 𝐵 No )
5 simplr 767 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → 0s <s 𝐴)
6 simprr 771 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → 0s <s 𝐵)
72, 3, 2, 4, 5, 6sltmuld 27522 . 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 27526 . . . . 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 27526 . . . . . 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 7412 . . 3 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = ( 0s -s 0s ))
14 subsid 27466 . . . 4 ( 0s No → ( 0s -s 0s ) = 0s )
151, 14ax-mp 5 . . 3 ( 0s -s 0s ) = 0s
1613, 15eqtrdi 2788 . 2 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = 0s )
17 muls01 27497 . . . . 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 7410 . . 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 27519 . . . . 5 ((𝐴 No 𝐵 No ) → (𝐴 ·s 𝐵) ∈ No )
2120ad2ant2r 745 . . . 4 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → (𝐴 ·s 𝐵) ∈ No )
22 subsid1 27465 . . . 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 2772 . 2 (((𝐴 No ∧ 0s <s 𝐴) ∧ (𝐵 No ∧ 0s <s 𝐵)) → ((𝐴 ·s 𝐵) -s (𝐴 ·s 0s )) = (𝐴 ·s 𝐵))
257, 16, 243brtr3d 5173 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 396   = wceq 1541  wcel 2106   class class class wbr 5142  (class class class)co 7394   No csur 27072   <s cslt 27073   0s c0s 27252   -s csubs 27424   ·s cmuls 27491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-1o 8450  df-2o 8451  df-nadd 8650  df-no 27075  df-slt 27076  df-bday 27077  df-sle 27177  df-sslt 27212  df-scut 27214  df-0s 27254  df-made 27271  df-old 27272  df-left 27274  df-right 27275  df-norec 27351  df-norec2 27362  df-adds 27373  df-negs 27425  df-subs 27426  df-muls 27492
This theorem is referenced by:  mulsgt0d  27530
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