Theorem muls01 28491

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 28188	. . 3 ⊢ 0s ∈ No
2		mulsval 28488	. . 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 4308	. . . . . . . . . 10 ⊢ ¬ ∃𝑞 ∈ ∅ 𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))
5		left0s 28272	. . . . . . . . . . 11 ⊢ ( L ‘ 0s ) = ∅
6	5	rexeqi 3319	. . . . . . . . . 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 3091	. . . . . . 7 ⊢ ¬ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))
10	9	abf 4364	. . . . . 6 ⊢ {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} = ∅
11		rex0 4308	. . . . . . . . . 10 ⊢ ¬ ∃𝑠 ∈ ∅ 𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))
12		right0s 28273	. . . . . . . . . . 11 ⊢ ( R ‘ 0s ) = ∅
13	12	rexeqi 3319	. . . . . . . . . 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 3091	. . . . . . 7 ⊢ ¬ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))
17	16	abf 4364	. . . . . 6 ⊢ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} = ∅
18	10, 17	uneq12i 4113	. . . . 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 4344	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
20	18, 19	eqtri 2784	. . . 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 4308	. . . . . . . . . 10 ⊢ ¬ ∃𝑢 ∈ ∅ 𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))
22	12	rexeqi 3319	. . . . . . . . . 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 3091	. . . . . . 7 ⊢ ¬ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))
26	25	abf 4364	. . . . . 6 ⊢ {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} = ∅
27		rex0 4308	. . . . . . . . . 10 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))
28	5	rexeqi 3319	. . . . . . . . . 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 3091	. . . . . . 7 ⊢ ¬ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))
32	31	abf 4364	. . . . . 6 ⊢ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} = ∅
33	26, 32	uneq12i 4113	. . . . 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 2784	. . . 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 7430	. . 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 28186	. . 3 ⊢ 0s = (∅ |s ∅)
37	35, 36	eqtr4i 2787	. 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 2812	1 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145  {cab 2739  ∃wrex 3087   ∪ cun 3897  ∅c0 4279  ‘cfv 6537  (class class class)co 7418   No csur 27990   |s ccuts 28138   0_s c0s 28184   L cleft 28204   R cright 28205   +_s cadds 28338   -_s csubs 28399   ·_s cmuls 28485

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-1o 8469  df-2o 8470  df-no 27993  df-lts 27994  df-bday 27995  df-slts 28137  df-cuts 28139  df-0s 28186  df-made 28206  df-old 28207  df-left 28209  df-right 28210  df-norec2 28328  df-muls 28486

This theorem is used by:  mulsrid  28492  muls02  28520  mulsgt0  28523  mulsge0d  28525  lemuls1ad  28561  muls0ord  28564  precsexlem9  28594  precsexlem11  28596  n0mulscl  28724  eucliddivs  28755  n0seo  28800  pw2gt0divsd  28824  pw2ge0divsd  28825  pw2divsnegd  28828  pw2divs0d  28834  z12bdaylem1  28849