Theorem mulscan2d 28196

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 27826	. . . . 5 ⊢ 0s ∈ No
3		ltnegs 28062	. . . . 5 ⊢ ((𝐶 ∈ No ∧ 0s ∈ No ) → (𝐶 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐶)))
4	1, 2, 3	sylancl 592	. . . 4 ⊢ (𝜑 → (𝐶 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐶)))
5		neg0s 28043	. . . . 5 ⊢ ( -us ‘ 0s ) = 0s
6	5	breq1i 5086	. . . 4 ⊢ (( -us ‘ 0s ) <s ( -us ‘𝐶) ↔ 0s <s ( -us ‘𝐶))
7	4, 6	bitrdi 288	. . 3 ⊢ (𝜑 → (𝐶 <s 0s ↔ 0s <s ( -us ‘𝐶)))
8		mulscan2d.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ No )
9	8, 1	mulnegs2d 28178	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s ( -us ‘𝐶)) = ( -us ‘(𝐴 ·s 𝐶)))
10		mulscan2d.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ No )
11	10, 1	mulnegs2d 28178	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s ( -us ‘𝐶)) = ( -us ‘(𝐵 ·s 𝐶)))
12	9, 11	eqeq12d 2756	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ ( -us ‘(𝐴 ·s 𝐶)) = ( -us ‘(𝐵 ·s 𝐶))))
13	8, 1	mulscld 28152	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
14	10, 1	mulscld 28152	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ∈ No )
15		negs11 28066	. . . . . . 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 590	. . . . . 6 ⊢ (𝜑 → (( -us ‘(𝐴 ·s 𝐶)) = ( -us ‘(𝐵 ·s 𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
17	12, 16	bitrd 280	. . . . 5 ⊢ (𝜑 → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
18	17	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
19	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → 𝐴 ∈ No )
20	10	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → 𝐵 ∈ No )
21	1	negscld 28054	. . . . . 6 ⊢ (𝜑 → ( -us ‘𝐶) ∈ No )
22	21	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ( -us ‘𝐶) ∈ No )
23		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → 0s <s ( -us ‘𝐶))
24	19, 20, 22, 23	mulscan2dlem 28195	. . . 4 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ 𝐴 = 𝐵))
25	18, 24	bitr3d 282	. . 3 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
26	7, 25	sylbida 598	. 2 ⊢ ((𝜑 ∧ 𝐶 <s 0s ) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
27	8	adantr 481	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐴 ∈ No )
28	10	adantr 481	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐵 ∈ No )
29	1	adantr 481	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐶 ∈ No )
30		simpr 485	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 0s <s 𝐶)
31	27, 28, 29, 30	mulscan2dlem 28195	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐶) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
32		mulscan2d.4	. . 3 ⊢ (𝜑 → 𝐶 ≠ 0s )
33		ltstrine 27740	. . . 4 ⊢ ((𝐶 ∈ No ∧ 0s ∈ No ) → (𝐶 ≠ 0s ↔ (𝐶 <s 0s ∨ 0s <s 𝐶)))
34	1, 2, 33	sylancl 592	. . 3 ⊢ (𝜑 → (𝐶 ≠ 0s ↔ (𝐶 <s 0s ∨ 0s <s 𝐶)))
35	32, 34	mpbid 233	. 2 ⊢ (𝜑 → (𝐶 <s 0s ∨ 0s <s 𝐶))
36	26, 31, 35	mpjaodan 966	1 ⊢ (𝜑 → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2935   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363   No csur 27628   <s clts 27629   0_s c0s 27822   -u_s cnegs 28036   ·_s cmuls 28123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-1o 8402  df-2o 8403  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038  df-subs 28039  df-muls 28124

This theorem is referenced by:  mulscan1d  28197  muls0ord  28202