Theorem mulsgt0 28337

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		0no 28002	. . . 4 ⊢ 0s ∈ No
2	1	a1i 11	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s ∈ No )
3		simpll 778	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 𝐴 ∈ No )
4		simprl 782	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 𝐵 ∈ No )
5		simplr 780	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s 𝐴)
6		simprr 784	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s 𝐵)
7	2, 3, 2, 4, 5, 6	ltmulsd 28330	. 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 28334	. . . . 5 ⊢ (𝐵 ∈ No → ( 0s ·s 𝐵) = 0s )
9	4, 8	syl 18	. . . 4 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → ( 0s ·s 𝐵) = 0s )
10		muls02 28334	. . . . . 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 7428	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = ( 0s -s 0s ))
14		subsid 28262	. . . 4 ⊢ ( 0s ∈ No → ( 0s -s 0s ) = 0s )
15	1, 14	ax-mp 5	. . 3 ⊢ ( 0s -s 0s ) = 0s
16	13, 15	eqtrdi 2814	. 2 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (( 0s ·s 𝐵) -s ( 0s ·s 0s )) = 0s )
17		muls01 28305	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
18	3, 17	syl 18	. . . 4 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (𝐴 ·s 0s ) = 0s )
19	18	oveq2d 7426	. . 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 28327	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) ∈ No )
21	20	ad2ant2r 759	. . . 4 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → (𝐴 ·s 𝐵) ∈ No )
22		subsid1 28261	. . . 4 ⊢ ((𝐴 ·s 𝐵) ∈ No → ((𝐴 ·s 𝐵) -s 0s ) = (𝐴 ·s 𝐵))
23	21, 22	syl 18	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → ((𝐴 ·s 𝐵) -s 0s ) = (𝐴 ·s 𝐵))
24	19, 23	eqtrd 2798	. 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 5142	1 ⊢ (((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s (𝐴 ·s 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   class class class wbr 5109  (class class class)co 7410   No csur 27804   <s clts 27805   0_s c0s 27998   -_s csubs 28213   ·_s cmuls 28299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-1o 8449  df-2o 8450  df-nadd 8648  df-no 27807  df-lts 27808  df-bday 27809  df-les 27909  df-slts 27951  df-cuts 27953  df-0s 28000  df-made 28020  df-old 28021  df-left 28023  df-right 28024  df-norec 28131  df-norec2 28142  df-adds 28153  df-negs 28214  df-subs 28215  df-muls 28300

This theorem is referenced by:  mulsgt0d  28338