Theorem muls01 28555

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		0no 28252	. . 3 ⊢ 0s ∈ No
2		mulsval 28552	. . 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 704	. 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 4311	. . . . . . . . . 10 ⊢ ¬ ∃𝑞 ∈ ∅ 𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))
5		left0s 28336	. . . . . . . . . . 11 ⊢ (L‘ 0s ) = ∅
6	5	rexeqi 3322	. . . . . . . . . 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 326	. . . . . . . . 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 3094	. . . . . . 7 ⊢ ¬ ∃𝑝 ∈ (L‘𝐴)∃𝑞 ∈ (L‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))
10	9	abf 4367	. . . . . 6 ⊢ {𝑎 ∣ ∃𝑝 ∈ (L‘𝐴)∃𝑞 ∈ (L‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))} = ∅
11		rex0 4311	. . . . . . . . . 10 ⊢ ¬ ∃𝑠 ∈ ∅ 𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))
12		right0s 28337	. . . . . . . . . . 11 ⊢ (R‘ 0s ) = ∅
13	12	rexeqi 3322	. . . . . . . . . 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 326	. . . . . . . . 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 3094	. . . . . . 7 ⊢ ¬ ∃𝑟 ∈ (R‘𝐴)∃𝑠 ∈ (R‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))
17	16	abf 4367	. . . . . 6 ⊢ {𝑏 ∣ ∃𝑟 ∈ (R‘𝐴)∃𝑠 ∈ (R‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))} = ∅
18	10, 17	uneq12i 4168	. . . . 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 4347	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
20	18, 19	eqtri 2786	. . . 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 4311	. . . . . . . . . 10 ⊢ ¬ ∃𝑢 ∈ ∅ 𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))
22	12	rexeqi 3322	. . . . . . . . . 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 326	. . . . . . . . 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 3094	. . . . . . 7 ⊢ ¬ ∃𝑡 ∈ (L‘𝐴)∃𝑢 ∈ (R‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))
26	25	abf 4367	. . . . . 6 ⊢ {𝑐 ∣ ∃𝑡 ∈ (L‘𝐴)∃𝑢 ∈ (R‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} = ∅
27		rex0 4311	. . . . . . . . . 10 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))
28	5	rexeqi 3322	. . . . . . . . . 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 326	. . . . . . . . 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 3094	. . . . . . 7 ⊢ ¬ ∃𝑣 ∈ (R‘𝐴)∃𝑤 ∈ (L‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))
32	31	abf 4367	. . . . . 6 ⊢ {𝑑 ∣ ∃𝑣 ∈ (R‘𝐴)∃𝑤 ∈ (L‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))} = ∅
33	26, 32	uneq12i 4168	. . . . 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 2786	. . . 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 7463	. . 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 28250	. . 3 ⊢ 0s = (∅ |s ∅)
37	35, 36	eqtr4i 2789	. 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 2814	1 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090   ∪ cun 4152  ∅c0 4281  ‘cfv 6566  (class class class)co 7451  Nocsur 28054   |s ccuts 28202   0_s c0s 28248  Lcleft 28268  Rcright 28269   +_s cadds 28402   −_s csubs 28463   ·_s cmuls 28549

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-0s 28250  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec2 28392  df-muls 28550

This theorem is used by:  mulsrid  28556  muls02  28584  mulsgt0  28587  mulsge0d  28589  lemuls1ad  28625  muls0ord  28628  precsexlem9  28658  precsexlem11  28660  n0mulscl  28788  eucliddivs  28819  n0seo  28864  pw2gt0divsd  28888  pw2ge0divsd  28889  pw2divsnegd  28892  pw2divs0d  28898  z12bdaylem1  28913