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Theorem mulscan2d 28219
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 27885 . . . . 5 0s No
3 sltneg 28091 . . . . 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 28072 . . . . 5 ( -us ‘ 0s ) = 0s
65breq1i 5154 . . . 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 28201 . . . . . . 7 (𝜑 → (𝐴 ·s ( -us𝐶)) = ( -us ‘(𝐴 ·s 𝐶)))
10 mulscan2d.2 . . . . . . . 8 (𝜑𝐵 No )
1110, 1mulnegs2d 28201 . . . . . . 7 (𝜑 → (𝐵 ·s ( -us𝐶)) = ( -us ‘(𝐵 ·s 𝐶)))
129, 11eqeq12d 2750 . . . . . 6 (𝜑 → ((𝐴 ·s ( -us𝐶)) = (𝐵 ·s ( -us𝐶)) ↔ ( -us ‘(𝐴 ·s 𝐶)) = ( -us ‘(𝐵 ·s 𝐶))))
138, 1mulscld 28175 . . . . . . 7 (𝜑 → (𝐴 ·s 𝐶) ∈ No )
1410, 1mulscld 28175 . . . . . . 7 (𝜑 → (𝐵 ·s 𝐶) ∈ No )
15 negs11 28095 . . . . . . 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 28083 . . . . . 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 28218 . . . 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 28218 . 2 ((𝜑 ∧ 0s <s 𝐶) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
32 mulscan2d.4 . . 3 (𝜑𝐶 ≠ 0s )
33 slttrine 27810 . . . 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 960 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 847   = wceq 1536  wcel 2105  wne 2937   class class class wbr 5147  cfv 6562  (class class class)co 7430   No csur 27698   <s cslt 27699   0s c0s 27881   -us cnegs 28065   ·s cmuls 28146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-ot 4639  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-1o 8504  df-2o 8505  df-nadd 8702  df-no 27701  df-slt 27702  df-bday 27703  df-sle 27804  df-sslt 27840  df-scut 27842  df-0s 27883  df-made 27900  df-old 27901  df-left 27903  df-right 27904  df-norec 27985  df-norec2 27996  df-adds 28007  df-negs 28067  df-subs 28068  df-muls 28147
This theorem is referenced by:  mulscan1d  28220  muls0ord  28225
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