Theorem mulscan2d 28185

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 27815	. . . . 5 ⊢ 0s ∈ No
3		ltnegs 28051	. . . . 5 ⊢ ((𝐶 ∈ No ∧ 0s ∈ No ) → (𝐶 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐶)))
4	1, 2, 3	sylancl 587	. . . 4 ⊢ (𝜑 → (𝐶 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐶)))
5		neg0s 28032	. . . . 5 ⊢ ( -us ‘ 0s ) = 0s
6	5	breq1i 5093	. . . 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 28167	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s ( -us ‘𝐶)) = ( -us ‘(𝐴 ·s 𝐶)))
10		mulscan2d.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ No )
11	10, 1	mulnegs2d 28167	. . . . . . 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 28141	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
14	10, 1	mulscld 28141	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·s 𝐶) ∈ No )
15		negs11 28055	. . . . . . 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 585	. . . . . 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 28043	. . . . . 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 28184	. . . 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 593	. 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 28184	. 2 ⊢ ((𝜑 ∧ 0s <s 𝐶) → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
32		mulscan2d.4	. . 3 ⊢ (𝜑 → 𝐶 ≠ 0s )
33		ltstrine 27729	. . . 4 ⊢ ((𝐶 ∈ No ∧ 0s ∈ No ) → (𝐶 ≠ 0s ↔ (𝐶 <s 0s ∨ 0s <s 𝐶)))
34	1, 2, 33	sylancl 587	. . 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 1542   ∈ wcel 2114   ≠ wne 2933   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360   No csur 27617   <s clts 27618   0_s c0s 27811   -u_s cnegs 28025   ·_s cmuls 28112

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-1o 8398  df-2o 8399  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-les 27723  df-slts 27764  df-cuts 27766  df-0s 27813  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec 27944  df-norec2 27955  df-adds 27966  df-negs 28027  df-subs 28028  df-muls 28113

This theorem is referenced by:  mulscan1d  28186  muls0ord  28191