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	⊢ (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		0sno 27871	. . . . 5 ⊢ 0s ∈ No
3		sltneg 28077	. . . . 5 ⊢ ((𝐶 ∈ No ∧ 0s ∈ No ) → (𝐶 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐶)))
4	1, 2, 3	sylancl 586	. . . 4 ⊢ (𝜑 → (𝐶 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐶)))
5		negs0s 28058	. . . . 5 ⊢ ( -us ‘ 0s ) = 0s
6	5	breq1i 5150	. . . 4 ⊢ (( -us ‘ 0s ) <s ( -us ‘𝐶) ↔ 0s <s ( -us ‘𝐶))
7	4, 6	bitrdi 287	. . 3 ⊢ (𝜑 → (𝐶 <s 0s ↔ 0s <s ( -us ‘𝐶)))
8		mulscan2d.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ No )
9	8, 1	mulnegs2d 28187	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s ( -us ‘𝐶)) = ( -us ‘(𝐴 ·s 𝐶)))
10		mulscan2d.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ No )
11	10, 1	mulnegs2d 28187	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s ( -us ‘𝐶)) = ( -us ‘(𝐵 ·s 𝐶)))
12	9, 11	eqeq12d 2753	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ ( -us ‘(𝐴 ·s 𝐶)) = ( -us ‘(𝐵 ·s 𝐶))))
13	8, 1	mulscld 28161	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
14	10, 1	mulscld 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)))
16	13, 14, 15	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (( -us ‘(𝐴 ·s 𝐶)) = ( -us ‘(𝐵 ·s 𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
17	12, 16	bitrd 279	. . . . 5 ⊢ (𝜑 → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
18	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
19	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → 𝐴 ∈ No )
20	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → 𝐵 ∈ No )
21	1	negscld 28069	. . . . . 6 ⊢ (𝜑 → ( -us ‘𝐶) ∈ No )
22	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ( -us ‘𝐶) ∈ No )
23		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → 0s <s ( -us ‘𝐶))
24	19, 20, 22, 23	mulscan2dlem 28204	. . . 4 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ 𝐴 = 𝐵))
25	18, 24	bitr3d 281	. . 3 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
26	7, 25	sylbida 592	. 2 ⊢ ((𝜑 ∧ 𝐶 <s 0s ) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
27	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐴 ∈ No )
28	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐵 ∈ No )
29	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐶 ∈ No )
30		simpr 484	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 0s <s 𝐶)
31	27, 28, 29, 30	mulscan2dlem 28204	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐶) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
32		mulscan2d.4	. . 3 ⊢ (𝜑 → 𝐶 ≠ 0s )
33		slttrine 27796	. . . 4 ⊢ ((𝐶 ∈ No ∧ 0s ∈ No ) → (𝐶 ≠ 0s ↔ (𝐶 <s 0s ∨ 0s <s 𝐶)))
34	1, 2, 33	sylancl 586	. . 3 ⊢ (𝜑 → (𝐶 ≠ 0s ↔ (𝐶 <s 0s ∨ 0s <s 𝐶)))
35	32, 34	mpbid 232	. 2 ⊢ (𝜑 → (𝐶 <s 0s ∨ 0s <s 𝐶))
36	26, 31, 35	mpjaodan 961	1 ⊢ (𝜑 → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))

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   0_s c0s 27867   -u_s 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