Theorem mulscan2d 28598

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 28228	. . . . 5 ⊢ 0s ∈ No
3		ltnegs 28464	. . . . 5 ⊢ ((𝐶 ∈ No ∧ 0s ∈ No) → (𝐶 <s 0s ↔ ( -s‘ 0s ) <s ( -s‘𝐶)))
4	1, 2, 3	sylancl 598	. . . 4 ⊢ (𝜑 → (𝐶 <s 0s ↔ ( -s‘ 0s ) <s ( -s‘𝐶)))
5		neg0s 28445	. . . . 5 ⊢ ( -s‘ 0s ) = 0s
6	5	breq1i 5113	. . . 4 ⊢ (( -s‘ 0s ) <s ( -s‘𝐶) ↔ 0s <s ( -s‘𝐶))
7	4, 6	bitrdi 290	. . 3 ⊢ (𝜑 → (𝐶 <s 0s ↔ 0s <s ( -s‘𝐶)))
8		mulscan2d.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ No)
9	8, 1	mulnegs2d 28580	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s ( -s‘𝐶)) = ( -s‘(𝐴 ·s 𝐶)))
10		mulscan2d.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ No)
11	10, 1	mulnegs2d 28580	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s ( -s‘𝐶)) = ( -s‘(𝐵 ·s 𝐶)))
12	9, 11	eqeq12d 2779	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·s ( -s‘𝐶)) = (𝐵 ·s ( -s‘𝐶)) ↔ ( -s‘(𝐴 ·s 𝐶)) = ( -s‘(𝐵 ·s 𝐶))))
13	8, 1	mulscld 28554	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No)
14	10, 1	mulscld 28554	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ∈ No)
15		negs11 28468	. . . . . . 7 ⊢ (((𝐴 ·s 𝐶) ∈ No ∧ (𝐵 ·s 𝐶) ∈ No) → (( -s‘(𝐴 ·s 𝐶)) = ( -s‘(𝐵 ·s 𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
16	13, 14, 15	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (( -s‘(𝐴 ·s 𝐶)) = ( -s‘(𝐵 ·s 𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
17	12, 16	bitrd 282	. . . . 5 ⊢ (𝜑 → ((𝐴 ·s ( -s‘𝐶)) = (𝐵 ·s ( -s‘𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
18	17	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 0s <s ( -s‘𝐶)) → ((𝐴 ·s ( -s‘𝐶)) = (𝐵 ·s ( -s‘𝐶)) ↔ (𝐴 ·s 𝐶) = (𝐵 ·s 𝐶)))
19	8	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -s‘𝐶)) → 𝐴 ∈ No)
20	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -s‘𝐶)) → 𝐵 ∈ No)
21	1	negscld 28456	. . . . . 6 ⊢ (𝜑 → ( -s‘𝐶) ∈ No)
22	21	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -s‘𝐶)) → ( -s‘𝐶) ∈ No)
23		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 0s <s ( -s‘𝐶)) → 0s <s ( -s‘𝐶))
24	19, 20, 22, 23	mulscan2dlem 28597	. . . 4 ⊢ ((𝜑 ∧ 0s <s ( -s‘𝐶)) → ((𝐴 ·s ( -s‘𝐶)) = (𝐵 ·s ( -s‘𝐶)) ↔ 𝐴 = 𝐵))
25	18, 24	bitr3d 284	. . 3 ⊢ ((𝜑 ∧ 0s <s ( -s‘𝐶)) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
26	7, 25	sylbida 604	. 2 ⊢ ((𝜑 ∧ 𝐶 <s 0s ) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
27	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐴 ∈ No)
28	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐵 ∈ No)
29	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 𝐶 ∈ No)
30		simpr 490	. . 3 ⊢ ((𝜑 ∧ 0s <s 𝐶) → 0s <s 𝐶)
31	27, 28, 29, 30	mulscan2dlem 28597	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐶) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
32		mulscan2d.4	. . 3 ⊢ (𝜑 → 𝐶 ≠ 0s )
33		ltstrine 28141	. . . 4 ⊢ ((𝐶 ∈ No ∧ 0s ∈ No) → (𝐶 ≠ 0s ↔ (𝐶 <s 0s ∨ 0s <s 𝐶)))
34	1, 2, 33	sylancl 598	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446  Nocsur 28030   <s clts 28031   0_s c0s 28224   -_scnegs 28438   ·_s cmuls 28525

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-nadd 8706  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-0s 28226  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec 28357  df-norec2 28368  df-adds 28379  df-negs 28440  df-subs 28441  df-muls 28526

This theorem is used by:  mulscan1d  28599  muls0ord  28604