Theorem muls01 28120

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 27817	. . 3 ⊢ 0s ∈ No
2		mulsval 28117	. . 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 692	. 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 4314	. . . . . . . . . 10 ⊢ ¬ ∃𝑞 ∈ ∅ 𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))
5		left0s 27901	. . . . . . . . . . 11 ⊢ ( L ‘ 0s ) = ∅
6	5	rexeqi 3297	. . . . . . . . . 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 3066	. . . . . . 7 ⊢ ¬ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))
10	9	abf 4360	. . . . . 6 ⊢ {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘ 0s )𝑎 = (((𝑝 ·s 0s ) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} = ∅
11		rex0 4314	. . . . . . . . . 10 ⊢ ¬ ∃𝑠 ∈ ∅ 𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))
12		right0s 27902	. . . . . . . . . . 11 ⊢ ( R ‘ 0s ) = ∅
13	12	rexeqi 3297	. . . . . . . . . 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 3066	. . . . . . 7 ⊢ ¬ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))
17	16	abf 4360	. . . . . 6 ⊢ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘ 0s )𝑏 = (((𝑟 ·s 0s ) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} = ∅
18	10, 17	uneq12i 4120	. . . . 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 4348	. . . . 5 ⊢ (∅ ∪ ∅) = ∅
20	18, 19	eqtri 2760	. . . 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 4314	. . . . . . . . . 10 ⊢ ¬ ∃𝑢 ∈ ∅ 𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))
22	12	rexeqi 3297	. . . . . . . . . 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 3066	. . . . . . 7 ⊢ ¬ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))
26	25	abf 4360	. . . . . 6 ⊢ {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘ 0s )𝑐 = (((𝑡 ·s 0s ) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} = ∅
27		rex0 4314	. . . . . . . . . 10 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))
28	5	rexeqi 3297	. . . . . . . . . 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 3066	. . . . . . 7 ⊢ ¬ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))
32	31	abf 4360	. . . . . 6 ⊢ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘ 0s )𝑑 = (((𝑣 ·s 0s ) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} = ∅
33	26, 32	uneq12i 4120	. . . . 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 2760	. . . 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 7380	. . 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 27815	. . 3 ⊢ 0s = (∅ |s ∅)
37	35, 36	eqtr4i 2763	. 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 2788	1 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062   ∪ cun 3901  ∅c0 4287  ‘cfv 6500  (class class class)co 7368   No csur 27619   |s ccuts 27767   0_s c0s 27813   L cleft 27833   R cright 27834   +_s cadds 27967   -_s csubs 28028   ·_s cmuls 28114

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-1o 8407  df-2o 8408  df-no 27622  df-lts 27623  df-bday 27624  df-slts 27766  df-cuts 27768  df-0s 27815  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec2 27957  df-muls 28115

This theorem is referenced by:  mulsrid  28121  muls02  28149  mulsgt0  28152  mulsge0d  28154  lemuls1ad  28190  muls0ord  28193  precsexlem9  28223  precsexlem11  28225  n0mulscl  28353  eucliddivs  28384  n0seo  28429  pw2gt0divsd  28453  pw2ge0divsd  28454  pw2divsnegd  28457  pw2divs0d  28463  z12bdaylem1  28478