Theorem mulscan2d 28326

Description: Cancellation of surreal multiplication when the right term is nonzero. (Contributed by Scott Fenton, 10-Mar-2025.)

Hypotheses

Ref	Expression
mulscan2d.1	⊢ (𝜑 → 𝐴 ∈ No )
mulscan2d.2	⊢ (𝜑 → 𝐵 ∈ No )
mulscan2d.3	⊢ (𝜑 → 𝐶 ∈ No )
mulscan2d.4	⊢ (𝜑 → 𝐶 ≠ 0s )

Assertion

Ref	Expression
mulscan2d	⊢ (𝜑 → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))

Proof of Theorem mulscan2d

Step	Hyp	Ref	Expression
1		mulscan2d.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ No )
2		0no 27956	. . . . 5 ⊢ 0s ∈ No
3		ltnegs 28192	. . . . 5 ⊢ ((𝐶 ∈ No ∧ 0s ∈ No ) → (𝐶 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐶)))
4	1, 2, 3	sylancl 597	. . . 4 ⊢ (𝜑 → (𝐶 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐶)))
5		neg0s 28173	. . . . 5 ⊢ ( -us ‘ 0s ) = 0s
6	5	breq1i 5111	. . . 4 ⊢ (( -us ‘ 0s ) <s ( -us ‘𝐶) ↔ 0s <s ( -us ‘𝐶))
7	4, 6	bitrdi 290	. . 3 ⊢ (𝜑 → (𝐶 <s 0s ↔ 0s <s ( -us ‘𝐶)))
8		mulscan2d.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ No )
9	8, 1	mulnegs2d 28308	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s ( -us ‘𝐶)) = ( -us ‘(𝐴 ·s 𝐶)))
10		mulscan2d.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ No )
11	10, 1	mulnegs2d 28308	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s ( -us ‘𝐶)) = ( -us ‘(𝐵 ·s 𝐶)))
12	9, 11	eqeq12d 2781	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ ( -us ‘(𝐴 ·s 𝐶)) = ( -us ‘(𝐵 ·s 𝐶))))
13	8, 1	mulscld 28282	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
14	10, 1	mulscld 28282	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ∈ No )
15		negs11 28196	. . . . . . 7 ⊢ (((𝐴 ·s 𝐶) ∈ No ∧ (𝐵 ·s 𝐶) ∈ No ) → (( -us ‘(𝐴 ·s 𝐶)) = ( -us ‘(𝐵 ·s 𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
16	13, 14, 15	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (( -us ‘(𝐴 ·s 𝐶)) = ( -us ‘(𝐵 ·s 𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
17	12, 16	bitrd 282	. . . . 5 ⊢ (𝜑 → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
18	17	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
19	8	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → 𝐴 ∈ No )
20	10	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → 𝐵 ∈ No )
21	1	negscld 28184	. . . . . 6 ⊢ (𝜑 → ( -us ‘𝐶) ∈ No )
22	21	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ( -us ‘𝐶) ∈ No )
23		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → 0s <s ( -us ‘𝐶))
24	19, 20, 22, 23	mulscan2dlem 28325	. . . 4 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ 𝐴 = 𝐵))
25	18, 24	bitr3d 284	. . 3 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
26	7, 25	sylbida 603	. 2 ⊢ ((𝜑 ∧ 𝐶 <s 0s ) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
27	8	adantr 485	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐴 ∈ No )
28	10	adantr 485	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐵 ∈ No )
29	1	adantr 485	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐶 ∈ No )
30		simpr 489	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 0s <s 𝐶)
31	27, 28, 29, 30	mulscan2dlem 28325	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐶) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
32		mulscan2d.4	. . 3 ⊢ (𝜑 → 𝐶 ≠ 0s )
33		ltstrine 27869	. . . 4 ⊢ ((𝐶 ∈ No ∧ 0s ∈ No ) → (𝐶 ≠ 0s ↔ (𝐶 <s 0s ∨ 0s <s 𝐶)))
34	1, 2, 33	sylancl 597	. . 3 ⊢ (𝜑 → (𝐶 ≠ 0s ↔ (𝐶 <s 0s ∨ 0s <s 𝐶)))
35	32, 34	mpbid 235	. 2 ⊢ (𝜑 → (𝐶 <s 0s ∨ 0s <s 𝐶))
36	26, 31, 35	mpjaodan 973	1 ⊢ (𝜑 → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1563   ∈ wcel 2145   ≠ wne 2960   class class class wbr 5104  ‘cfv 6525  (class class class)co 7400   No csur 27758   <s clts 27759   0_s c0s 27952   -u_s cnegs 28166   ·_s cmuls 28253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4908  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-1o 8441  df-2o 8442  df-nadd 8640  df-no 27761  df-lts 27762  df-bday 27763  df-les 27863  df-slts 27905  df-cuts 27907  df-0s 27954  df-made 27974  df-old 27975  df-left 27977  df-right 27978  df-norec 28085  df-norec2 28096  df-adds 28107  df-negs 28168  df-subs 28169  df-muls 28254

This theorem is referenced by:  mulscan1d  28327  muls0ord  28332