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Theorem mulscan2d 28205
Description: Cancellation of surreal multiplication when the right term is non-zero. (Contributed by Scott Fenton, 10-Mar-2025.)
Hypotheses
Ref Expression
mulscan2d.1 (𝜑𝐴 No )
mulscan2d.2 (𝜑𝐵 No )
mulscan2d.3 (𝜑𝐶 No )
mulscan2d.4 (𝜑𝐶 ≠ 0s )
Assertion
Ref Expression
mulscan2d (𝜑 → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))

Proof of Theorem mulscan2d
StepHypRef Expression
1 mulscan2d.3 . . . . 5 (𝜑𝐶 No )
2 0sno 27871 . . . . 5 0s No
3 sltneg 28077 . . . . 5 ((𝐶 No ∧ 0s No ) → (𝐶 <s 0s ↔ ( -us ‘ 0s ) <s ( -us𝐶)))
41, 2, 3sylancl 586 . . . 4 (𝜑 → (𝐶 <s 0s ↔ ( -us ‘ 0s ) <s ( -us𝐶)))
5 negs0s 28058 . . . . 5 ( -us ‘ 0s ) = 0s
65breq1i 5150 . . . 4 (( -us ‘ 0s ) <s ( -us𝐶) ↔ 0s <s ( -us𝐶))
74, 6bitrdi 287 . . 3 (𝜑 → (𝐶 <s 0s ↔ 0s <s ( -us𝐶)))
8 mulscan2d.1 . . . . . . . 8 (𝜑𝐴 No )
98, 1mulnegs2d 28187 . . . . . . 7 (𝜑 → (𝐴 ·s ( -us𝐶)) = ( -us ‘(𝐴 ·s 𝐶)))
10 mulscan2d.2 . . . . . . . 8 (𝜑𝐵 No )
1110, 1mulnegs2d 28187 . . . . . . 7 (𝜑 → (𝐵 ·s ( -us𝐶)) = ( -us ‘(𝐵 ·s 𝐶)))
129, 11eqeq12d 2753 . . . . . 6 (𝜑 → ((𝐴 ·s ( -us𝐶)) = (𝐵 ·s ( -us𝐶)) ↔ ( -us ‘(𝐴 ·s 𝐶)) = ( -us ‘(𝐵 ·s 𝐶))))
138, 1mulscld 28161 . . . . . . 7 (𝜑 → (𝐴 ·s 𝐶) ∈ No )
1410, 1mulscld 28161 . . . . . . 7 (𝜑 → (𝐵 ·s 𝐶) ∈ No )
15 negs11 28081 . . . . . . 7 (((𝐴 ·s 𝐶) ∈ No ∧ (𝐵 ·s 𝐶) ∈ No ) → (( -us ‘(𝐴 ·s 𝐶)) = ( -us ‘(𝐵 ·s 𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
1613, 14, 15syl2anc 584 . . . . . 6 (𝜑 → (( -us ‘(𝐴 ·s 𝐶)) = ( -us ‘(𝐵 ·s 𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
1712, 16bitrd 279 . . . . 5 (𝜑 → ((𝐴 ·s ( -us𝐶)) = (𝐵 ·s ( -us𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
1817adantr 480 . . . 4 ((𝜑 ∧ 0s <s ( -us𝐶)) → ((𝐴 ·s ( -us𝐶)) = (𝐵 ·s ( -us𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
198adantr 480 . . . . 5 ((𝜑 ∧ 0s <s ( -us𝐶)) → 𝐴 No )
2010adantr 480 . . . . 5 ((𝜑 ∧ 0s <s ( -us𝐶)) → 𝐵 No )
211negscld 28069 . . . . . 6 (𝜑 → ( -us𝐶) ∈ No )
2221adantr 480 . . . . 5 ((𝜑 ∧ 0s <s ( -us𝐶)) → ( -us𝐶) ∈ No )
23 simpr 484 . . . . 5 ((𝜑 ∧ 0s <s ( -us𝐶)) → 0s <s ( -us𝐶))
2419, 20, 22, 23mulscan2dlem 28204 . . . 4 ((𝜑 ∧ 0s <s ( -us𝐶)) → ((𝐴 ·s ( -us𝐶)) = (𝐵 ·s ( -us𝐶)) ↔ 𝐴 = 𝐵))
2518, 24bitr3d 281 . . 3 ((𝜑 ∧ 0s <s ( -us𝐶)) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
267, 25sylbida 592 . 2 ((𝜑𝐶 <s 0s ) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
278adantr 480 . . 3 ((𝜑 ∧ 0s <s 𝐶) → 𝐴 No )
2810adantr 480 . . 3 ((𝜑 ∧ 0s <s 𝐶) → 𝐵 No )
291adantr 480 . . 3 ((𝜑 ∧ 0s <s 𝐶) → 𝐶 No )
30 simpr 484 . . 3 ((𝜑 ∧ 0s <s 𝐶) → 0s <s 𝐶)
3127, 28, 29, 30mulscan2dlem 28204 . 2 ((𝜑 ∧ 0s <s 𝐶) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
32 mulscan2d.4 . . 3 (𝜑𝐶 ≠ 0s )
33 slttrine 27796 . . . 4 ((𝐶 No ∧ 0s No ) → (𝐶 ≠ 0s ↔ (𝐶 <s 0s ∨ 0s <s 𝐶)))
341, 2, 33sylancl 586 . . 3 (𝜑 → (𝐶 ≠ 0s ↔ (𝐶 <s 0s ∨ 0s <s 𝐶)))
3532, 34mpbid 232 . 2 (𝜑 → (𝐶 <s 0s ∨ 0s <s 𝐶))
3626, 31, 35mpjaodan 961 1 (𝜑 → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561  (class class class)co 7431   No csur 27684   <s cslt 27685   0s c0s 27867   -us cnegs 28051   ·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:  mulscan1d  28206  muls0ord  28211
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