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Theorem sltmul2 28197
Description: Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025.)
Assertion
Ref Expression
sltmul2 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → (𝐵 <s 𝐶 ↔ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)))

Proof of Theorem sltmul2
StepHypRef Expression
1 simpl1l 1225 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 𝐴 No )
2 simpl3 1194 . . . . . 6 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 𝐶 No )
3 simpl2 1193 . . . . . 6 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 𝐵 No )
42, 3subscld 28093 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → (𝐶 -s 𝐵) ∈ No )
5 simpl1r 1226 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 0s <s 𝐴)
6 simp2 1138 . . . . . . 7 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → 𝐵 No )
7 simp3 1139 . . . . . . 7 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → 𝐶 No )
86, 7posdifsd 28127 . . . . . 6 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → (𝐵 <s 𝐶 ↔ 0s <s (𝐶 -s 𝐵)))
98biimpa 476 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 0s <s (𝐶 -s 𝐵))
101, 4, 5, 9mulsgt0d 28171 . . . 4 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 0s <s (𝐴 ·s (𝐶 -s 𝐵)))
111, 2, 3subsdid 28184 . . . 4 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s (𝐶 -s 𝐵)) = ((𝐴 ·s 𝐶) -s (𝐴 ·s 𝐵)))
1210, 11breqtrd 5169 . . 3 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 0s <s ((𝐴 ·s 𝐶) -s (𝐴 ·s 𝐵)))
131, 3mulscld 28161 . . . 4 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐵) ∈ No )
141, 2mulscld 28161 . . . 4 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐶) ∈ No )
1513, 14posdifsd 28127 . . 3 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → ((𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) ↔ 0s <s ((𝐴 ·s 𝐶) -s (𝐴 ·s 𝐵))))
1612, 15mpbird 257 . 2 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶))
17 simp1l 1198 . . . . . . . 8 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → 𝐴 No )
1817, 7mulscld 28161 . . . . . . 7 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → (𝐴 ·s 𝐶) ∈ No )
19 sltirr 27791 . . . . . . 7 ((𝐴 ·s 𝐶) ∈ No → ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐶))
2018, 19syl 17 . . . . . 6 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐶))
21 oveq2 7439 . . . . . . . 8 (𝐵 = 𝐶 → (𝐴 ·s 𝐵) = (𝐴 ·s 𝐶))
2221breq1d 5153 . . . . . . 7 (𝐵 = 𝐶 → ((𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) ↔ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐶)))
2322notbid 318 . . . . . 6 (𝐵 = 𝐶 → (¬ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) ↔ ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐶)))
2420, 23syl5ibrcom 247 . . . . 5 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → (𝐵 = 𝐶 → ¬ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)))
2524con2d 134 . . . 4 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → ((𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) → ¬ 𝐵 = 𝐶))
2625imp 406 . . 3 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → ¬ 𝐵 = 𝐶)
2717, 6mulscld 28161 . . . . . . 7 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → (𝐴 ·s 𝐵) ∈ No )
28 sltasym 27793 . . . . . . 7 (((𝐴 ·s 𝐵) ∈ No ∧ (𝐴 ·s 𝐶) ∈ No ) → ((𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) → ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐵)))
2927, 18, 28syl2anc 584 . . . . . 6 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → ((𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) → ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐵)))
3029imp 406 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐵))
31 simpl1l 1225 . . . . . . . 8 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐴 No )
3231adantr 480 . . . . . . 7 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → 𝐴 No )
33 simpll2 1214 . . . . . . . 8 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → 𝐵 No )
34 simpll3 1215 . . . . . . . 8 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → 𝐶 No )
3533, 34subscld 28093 . . . . . . 7 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → (𝐵 -s 𝐶) ∈ No )
36 simpl1r 1226 . . . . . . . 8 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 0s <s 𝐴)
3736adantr 480 . . . . . . 7 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → 0s <s 𝐴)
38 simpr 484 . . . . . . 7 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → 0s <s (𝐵 -s 𝐶))
3932, 35, 37, 38mulsgt0d 28171 . . . . . 6 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → 0s <s (𝐴 ·s (𝐵 -s 𝐶)))
4032, 33, 34subsdid 28184 . . . . . . . 8 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → (𝐴 ·s (𝐵 -s 𝐶)) = ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝐶)))
4140breq2d 5155 . . . . . . 7 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → ( 0s <s (𝐴 ·s (𝐵 -s 𝐶)) ↔ 0s <s ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝐶))))
4218ad2antrr 726 . . . . . . . 8 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → (𝐴 ·s 𝐶) ∈ No )
4327ad2antrr 726 . . . . . . . 8 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → (𝐴 ·s 𝐵) ∈ No )
4442, 43posdifsd 28127 . . . . . . 7 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → ((𝐴 ·s 𝐶) <s (𝐴 ·s 𝐵) ↔ 0s <s ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝐶))))
4541, 44bitr4d 282 . . . . . 6 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → ( 0s <s (𝐴 ·s (𝐵 -s 𝐶)) ↔ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐵)))
4639, 45mpbid 232 . . . . 5 (((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) ∧ 0s <s (𝐵 -s 𝐶)) → (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐵))
4730, 46mtand 816 . . . 4 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → ¬ 0s <s (𝐵 -s 𝐶))
48 simpl3 1194 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐶 No )
49 simpl2 1193 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐵 No )
5048, 49posdifsd 28127 . . . 4 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → (𝐶 <s 𝐵 ↔ 0s <s (𝐵 -s 𝐶)))
5147, 50mtbird 325 . . 3 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → ¬ 𝐶 <s 𝐵)
52 sltlin 27794 . . . 4 ((𝐵 No 𝐶 No ) → (𝐵 <s 𝐶𝐵 = 𝐶𝐶 <s 𝐵))
5349, 48, 52syl2anc 584 . . 3 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → (𝐵 <s 𝐶𝐵 = 𝐶𝐶 <s 𝐵))
5426, 51, 53ecase23d 1475 . 2 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐵 <s 𝐶)
5516, 54impbida 801 1 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → (𝐵 <s 𝐶 ↔ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3o 1086  w3a 1087   = wceq 1540  wcel 2108   class class class wbr 5143  (class class class)co 7431   No csur 27684   <s cslt 27685   0s c0s 27867   -s csubs 28052   ·s cmuls 28132
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133
This theorem is referenced by:  sltmul2d  28198
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