Theorem muls01 28505

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 28202	. . 3 ⊢ 0s ∈ No
2		mulsval 28502	. . 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 4307	. . . . . . . . . 10 ⊢ ¬ ∃𝑞 ∈ ∅ 𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))
5		left0s 28286	. . . . . . . . . . 11 ⊢ (L‘ 0s ) = ∅
6	5	rexeqi 3318	. . . . . . . . . 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 3090	. . . . . . 7 ⊢ ¬ ∃𝑝 ∈ (L‘𝐴)∃𝑞 ∈ (L‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))
10	9	abf 4363	. . . . . 6 ⊢ {𝑎 ∣ ∃𝑝 ∈ (L‘𝐴)∃𝑞 ∈ (L‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))} = ∅
11		rex0 4307	. . . . . . . . . 10 ⊢ ¬ ∃𝑠 ∈ ∅ 𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))
12		right0s 28287	. . . . . . . . . . 11 ⊢ (R‘ 0s ) = ∅
13	12	rexeqi 3318	. . . . . . . . . 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 3090	. . . . . . 7 ⊢ ¬ ∃𝑟 ∈ (R‘𝐴)∃𝑠 ∈ (R‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))
17	16	abf 4363	. . . . . 6 ⊢ {𝑏 ∣ ∃𝑟 ∈ (R‘𝐴)∃𝑠 ∈ (R‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))} = ∅
18	10, 17	uneq12i 4164	. . . . 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 4343	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
20	18, 19	eqtri 2783	. . . 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 4307	. . . . . . . . . 10 ⊢ ¬ ∃𝑢 ∈ ∅ 𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))
22	12	rexeqi 3318	. . . . . . . . . 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 3090	. . . . . . 7 ⊢ ¬ ∃𝑡 ∈ (L‘𝐴)∃𝑢 ∈ (R‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))
26	25	abf 4363	. . . . . 6 ⊢ {𝑐 ∣ ∃𝑡 ∈ (L‘𝐴)∃𝑢 ∈ (R‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} = ∅
27		rex0 4307	. . . . . . . . . 10 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))
28	5	rexeqi 3318	. . . . . . . . . 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 3090	. . . . . . 7 ⊢ ¬ ∃𝑣 ∈ (R‘𝐴)∃𝑤 ∈ (L‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))
32	31	abf 4363	. . . . . 6 ⊢ {𝑑 ∣ ∃𝑣 ∈ (R‘𝐴)∃𝑤 ∈ (L‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))} = ∅
33	26, 32	uneq12i 4164	. . . . 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 2783	. . . 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 7434	. . 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 28200	. . 3 ⊢ 0s = (∅ |s ∅)
37	35, 36	eqtr4i 2786	. 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 2811	1 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086   ∪ cun 4148  ∅c0 4277  ‘cfv 6541  (class class class)co 7422  Nocsur 28004   |s ccuts 28152   0_s c0s 28198  Lcleft 28218  Rcright 28219   +_s cadds 28352   −_s csubs 28413   ·_s cmuls 28499

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-1o 8481  df-2o 8482  df-no 28007  df-lts 28008  df-bday 28009  df-slts 28151  df-cuts 28153  df-0s 28200  df-made 28220  df-old 28221  df-left 28223  df-right 28224  df-norec2 28342  df-muls 28500

This theorem is used by:  mulsrid  28506  muls02  28534  mulsgt0  28537  mulsge0d  28539  lemuls1ad  28575  muls0ord  28578  precsexlem9  28608  precsexlem11  28610  n0mulscl  28738  eucliddivs  28769  n0seo  28814  pw2gt0divsd  28838  pw2ge0divsd  28839  pw2divsnegd  28842  pw2divs0d  28848  z12bdaylem1  28863