Theorem mulscan2d 27560

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 27256	. . . . 5 ⊢ 0s ∈ No
3		sltneg 27448	. . . . 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 27430	. . . . 5 ⊢ ( -us ‘ 0s ) = 0s
6	5	breq1i 5149	. . . 4 ⊢ (( -us ‘ 0s ) <s ( -us ‘𝐶) ↔ 0s <s ( -us ‘𝐶))
7	4, 6	bitrdi 286	. . 3 ⊢ (𝜑 → (𝐶 <s 0s ↔ 0s <s ( -us ‘𝐶)))
8		mulscan2d.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ No )
9	8, 1	mulnegs2d 27545	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s ( -us ‘𝐶)) = ( -us ‘(𝐴 ·s 𝐶)))
10		mulscan2d.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ No )
11	10, 1	mulnegs2d 27545	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s ( -us ‘𝐶)) = ( -us ‘(𝐵 ·s 𝐶)))
12	9, 11	eqeq12d 2748	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ ( -us ‘(𝐴 ·s 𝐶)) = ( -us ‘(𝐵 ·s 𝐶))))
13	8, 1	mulscld 27520	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
14	10, 1	mulscld 27520	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ∈ No )
15		negs11 27452	. . . . . . 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 278	. . . . 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 27440	. . . . . 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 27559	. . . 4 ⊢ ((𝜑 ∧ 0s <s ( -us ‘𝐶)) → ((𝐴 ·s ( -us ‘𝐶)) = (𝐵 ·s ( -us ‘𝐶)) ↔ 𝐴 = 𝐵))
25	18, 24	bitr3d 280	. . 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 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 27559	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐶) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
32		mulscan2d.4	. . 3 ⊢ (𝜑 → 𝐶 ≠ 0s )
33		slttrine 27183	. . . 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 231	. 2 ⊢ (𝜑 → (𝐶 <s 0s ∨ 0s <s 𝐶))
36	26, 31, 35	mpjaodan 957	1 ⊢ (𝜑 → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   ≠ wne 2940   class class class wbr 5142  ‘cfv 6533  (class class class)co 7394   No csur 27072   <s cslt 27073   0_s c0s 27252   -u_s cnegs 27423   ·_s cmuls 27491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-1o 8450  df-2o 8451  df-nadd 8650  df-no 27075  df-slt 27076  df-bday 27077  df-sle 27177  df-sslt 27212  df-scut 27214  df-0s 27254  df-made 27271  df-old 27272  df-left 27274  df-right 27275  df-norec 27351  df-norec2 27362  df-adds 27373  df-negs 27425  df-subs 27426  df-muls 27492

This theorem is referenced by:  mulscan1d  27561