Theorem mulsgt0 28113

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

Step	Hyp	Ref	Expression
1		0sno 27797	. . . 4 ⊢ 0s ∈ No
2	1	a1i 11	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s ∈ No )
3		simpll 766	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 𝐴 ∈ No )
4		simprl 770	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 𝐵 ∈ No )
5		simplr 768	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s 𝐴)
6		simprr 772	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s 𝐵)
7	2, 3, 2, 4, 5, 6	sltmuld 28106	. 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 28110	. . . . 5 ⊢ (𝐵 ∈ No → ( 0s ·s 𝐵) = 0s )
9	4, 8	syl 17	. . . 4 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → ( 0s ·s 𝐵) = 0s )
10		muls02 28110	. . . . . 6 ⊢ ( 0s ∈ No → ( 0s ·s 0s ) = 0s )
11	1, 10	ax-mp 5	. . . . 5 ⊢ ( 0s ·s 0s ) = 0s
12	11	a1i 11	. . . 4 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → ( 0s ·s 0s ) = 0s )
13	9, 12	oveq12d 7374	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = ( 0s -s 0s ))
14		subsid 28038	. . . 4 ⊢ ( 0s ∈ No → ( 0s -s 0s ) = 0s )
15	1, 14	ax-mp 5	. . 3 ⊢ ( 0s -s 0s ) = 0s
16	13, 15	eqtrdi 2785	. 2 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = 0s )
17		muls01 28081	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
18	3, 17	syl 17	. . . 4 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (𝐴 ·s 0s ) = 0s )
19	18	oveq2d 7372	. . 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 28103	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) ∈ No )
21	20	ad2ant2r 747	. . . 4 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (𝐴 ·s 𝐵) ∈ No )
22		subsid1 28037	. . . 4 ⊢ ((𝐴 ·s 𝐵) ∈ No → ((𝐴 ·s 𝐵) -s 0s ) = (𝐴 ·s 𝐵))
23	21, 22	syl 17	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → ((𝐴 ·s 𝐵) -s 0s ) = (𝐴 ·s 𝐵))
24	19, 23	eqtrd 2769	. 2 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → ((𝐴 ·s 𝐵) -s (𝐴 ·s 0s )) = (𝐴 ·s 𝐵))
25	7, 16, 24	3brtr3d 5127	1 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s (𝐴 ·s 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   class class class wbr 5096  (class class class)co 7356   No csur 27605   <s cslt 27606   0_s c0s 27793   -_s csubs 27989   ·_s cmuls 28075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-ot 4587  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-nadd 8592  df-no 27608  df-slt 27609  df-bday 27610  df-sle 27711  df-sslt 27748  df-scut 27750  df-0s 27795  df-made 27815  df-old 27816  df-left 27818  df-right 27819  df-norec 27908  df-norec2 27919  df-adds 27930  df-negs 27990  df-subs 27991  df-muls 28076

This theorem is referenced by:  mulsgt0d  28114