Theorem mulsgt0 28023

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 27714	. . . 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 28016	. 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 28020	. . . . 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 28020	. . . . . 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 7387	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = ( 0s -s 0s ))
14		subsid 27949	. . . 4 ⊢ ( 0s ∈ No → ( 0s -s 0s ) = 0s )
15	1, 14	ax-mp 5	. . 3 ⊢ ( 0s -s 0s ) = 0s
16	13, 15	eqtrdi 2780	. 2 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = 0s )
17		muls01 27991	. . . . 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 7385	. . 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 28013	. . . . 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 27948	. . . 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 2764	. 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 5133	1 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s (𝐴 ·s 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   class class class wbr 5102  (class class class)co 7369   No csur 27527   <s cslt 27528   0_s c0s 27710   -_s csubs 27902   ·_s cmuls 27985

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-1o 8411  df-2o 8412  df-nadd 8607  df-no 27530  df-slt 27531  df-bday 27532  df-sle 27633  df-sslt 27669  df-scut 27671  df-0s 27712  df-made 27731  df-old 27732  df-left 27734  df-right 27735  df-norec 27821  df-norec2 27832  df-adds 27843  df-negs 27903  df-subs 27904  df-muls 27986

This theorem is referenced by:  mulsgt0d  28024