Theorem muls01 28138

Description: Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)

Assertion

Ref	Expression
muls01	⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )

Proof of Theorem muls01

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑝 𝑞 𝑟 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0sno 27871	. . 3 ⊢ 0s ∈ No
2		mulsval 28135	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ·s 0s ) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
3	1, 2	mpan2 691	. 2 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
4		rex0 4360	. . . . . . . . . 10 ⊢ ¬ ∃𝑞 ∈ ∅ 𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))
5		left0s 27931	. . . . . . . . . . 11 ⊢ ( L ‘ 0s ) = ∅
6	5	rexeqi 3325	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ ∃𝑞 ∈ ∅ 𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
7	4, 6	mtbir 323	. . . . . . . . 9 ⊢ ¬ ∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))
8	7	a1i 11	. . . . . . . 8 ⊢ (𝑝 ∈ ( L ‘𝐴) → ¬ ∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
9	8	nrex 3074	. . . . . . 7 ⊢ ¬ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))
10	9	abf 4406	. . . . . 6 ⊢ {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} = ∅
11		rex0 4360	. . . . . . . . . 10 ⊢ ¬ ∃𝑠 ∈ ∅ 𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))
12		right0s 27932	. . . . . . . . . . 11 ⊢ ( R ‘ 0s ) = ∅
13	12	rexeqi 3325	. . . . . . . . . 10 ⊢ (∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ↔ ∃𝑠 ∈ ∅ 𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
14	11, 13	mtbir 323	. . . . . . . . 9 ⊢ ¬ ∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))
15	14	a1i 11	. . . . . . . 8 ⊢ (𝑟 ∈ ( R ‘𝐴) → ¬ ∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
16	15	nrex 3074	. . . . . . 7 ⊢ ¬ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))
17	16	abf 4406	. . . . . 6 ⊢ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} = ∅
18	10, 17	uneq12i 4166	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) = (∅ ∪ ∅)
19		un0 4394	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
20	18, 19	eqtri 2765	. . . 4 ⊢ ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) = ∅
21		rex0 4360	. . . . . . . . . 10 ⊢ ¬ ∃𝑢 ∈ ∅ 𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))
22	12	rexeqi 3325	. . . . . . . . . 10 ⊢ (∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ↔ ∃𝑢 ∈ ∅ 𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
23	21, 22	mtbir 323	. . . . . . . . 9 ⊢ ¬ ∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))
24	23	a1i 11	. . . . . . . 8 ⊢ (𝑡 ∈ ( L ‘𝐴) → ¬ ∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
25	24	nrex 3074	. . . . . . 7 ⊢ ¬ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))
26	25	abf 4406	. . . . . 6 ⊢ {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} = ∅
27		rex0 4360	. . . . . . . . . 10 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))
28	5	rexeqi 3325	. . . . . . . . . 10 ⊢ (∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ ∃𝑤 ∈ ∅ 𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))
29	27, 28	mtbir 323	. . . . . . . . 9 ⊢ ¬ ∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))
30	29	a1i 11	. . . . . . . 8 ⊢ (𝑣 ∈ ( R ‘𝐴) → ¬ ∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))
31	30	nrex 3074	. . . . . . 7 ⊢ ¬ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))
32	31	abf 4406	. . . . . 6 ⊢ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} = ∅
33	26, 32	uneq12i 4166	. . . . 5 ⊢ ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) = (∅ ∪ ∅)
34	33, 19	eqtri 2765	. . . 4 ⊢ ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) = ∅
35	20, 34	oveq12i 7443	. . 3 ⊢ (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) = (∅ |s ∅)
36		df-0s 27869	. . 3 ⊢ 0s = (∅ |s ∅)
37	35, 36	eqtr4i 2768	. 2 ⊢ (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) = 0s
38	3, 37	eqtrdi 2793	1 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2108  {cab 2714  ∃wrex 3070   ∪ cun 3949  ∅c0 4333  ‘cfv 6561  (class class class)co 7431   No csur 27684   |s cscut 27827   0_s c0s 27867   L cleft 27884   R cright 27885   +_s cadds 27992   -_s csubs 28052   ·_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-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-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec2 27982  df-muls 28133

This theorem is referenced by:  mulsrid  28139  muls02  28167  mulsgt0  28170  mulsge0d  28172  slemul1ad  28208  muls0ord  28211  precsexlem9  28239  precsexlem11  28241  n0mulscl  28348  n0seo  28405  cutpw2  28417