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Theorem sltmul2 27552
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 1224 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 𝐴 No )
2 simpl3 1193 . . . . . 6 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 𝐶 No )
3 simpl2 1192 . . . . . 6 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 𝐵 No )
42, 3subscld 27464 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → (𝐶 -s 𝐵) ∈ No )
5 simpl1r 1225 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 0s <s 𝐴)
6 simp2 1137 . . . . . . 7 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → 𝐵 No )
7 simp3 1138 . . . . . . 7 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → 𝐶 No )
86, 7posdifsd 27490 . . . . . 6 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → (𝐵 <s 𝐶 ↔ 0s <s (𝐶 -s 𝐵)))
98biimpa 477 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 0s <s (𝐶 -s 𝐵))
101, 4, 5, 9mulsgt0d 27530 . . . 4 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 0s <s (𝐴 ·s (𝐶 -s 𝐵)))
111, 2, 3subsdid 27542 . . . 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 5168 . . 3 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → 0s <s ((𝐴 ·s 𝐶) -s (𝐴 ·s 𝐵)))
131, 3mulscld 27520 . . . 4 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐵) ∈ No )
141, 2mulscld 27520 . . . 4 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐶) ∈ No )
1513, 14posdifsd 27490 . . 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 256 . 2 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ 𝐵 <s 𝐶) → (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶))
17 simp1l 1197 . . . . . . . 8 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → 𝐴 No )
1817, 7mulscld 27520 . . . . . . 7 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → (𝐴 ·s 𝐶) ∈ No )
19 sltirr 27178 . . . . . . 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 7402 . . . . . . . 8 (𝐵 = 𝐶 → (𝐴 ·s 𝐵) = (𝐴 ·s 𝐶))
2221breq1d 5152 . . . . . . 7 (𝐵 = 𝐶 → ((𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) ↔ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐶)))
2322notbid 317 . . . . . 6 (𝐵 = 𝐶 → (¬ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶) ↔ ¬ (𝐴 ·s 𝐶) <s (𝐴 ·s 𝐶)))
2420, 23syl5ibrcom 246 . . . . 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 407 . . 3 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → ¬ 𝐵 = 𝐶)
2717, 6mulscld 27520 . . . . . . 7 (((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) → (𝐴 ·s 𝐵) ∈ No )
28 sltasym 27180 . . . . . . 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 407 . . . . 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 1224 . . . . . . . 8 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐴 No )
3231adantr 481 . . . . . . 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 1213 . . . . . . . 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 1214 . . . . . . . 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 27464 . . . . . . 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 1225 . . . . . . . 8 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 0s <s 𝐴)
3736adantr 481 . . . . . . 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 485 . . . . . . 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 27530 . . . . . 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 27542 . . . . . . . 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 5154 . . . . . . 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 724 . . . . . . . 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 724 . . . . . . . 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 27490 . . . . . . 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 281 . . . . . 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 231 . . . . 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 814 . . . 4 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → ¬ 0s <s (𝐵 -s 𝐶))
48 simpl3 1193 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐶 No )
49 simpl2 1192 . . . . 5 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐵 No )
5048, 49posdifsd 27490 . . . 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 324 . . 3 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → ¬ 𝐶 <s 𝐵)
52 sltlin 27181 . . . 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 1473 . 2 ((((𝐴 No ∧ 0s <s 𝐴) ∧ 𝐵 No 𝐶 No ) ∧ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)) → 𝐵 <s 𝐶)
5516, 54impbida 799 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 205  wa 396  w3o 1086  w3a 1087   = 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:  sltmul2d  27553
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