Theorem recsex 28243

Description: A non-zero surreal has a reciprocal. (Contributed by Scott Fenton, 15-Mar-2025.)

Assertion

Ref	Expression
recsex	⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )

Distinct variable group:   𝑥,𝐴

Proof of Theorem recsex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0sno 27871	. . . 4 ⊢ 0s ∈ No
2		slttrine 27796	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≠ 0s ↔ (𝐴 <s 0s ∨ 0s <s 𝐴)))
3	1, 2	mpan2 691	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ≠ 0s ↔ (𝐴 <s 0s ∨ 0s <s 𝐴)))
4		sltneg 28077	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐴)))
5	1, 4	mpan2 691	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐴)))
6		negs0s 28058	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
7	6	breq1i 5150	. . . . . 6 ⊢ (( -us ‘ 0s ) <s ( -us ‘𝐴) ↔ 0s <s ( -us ‘𝐴))
8	5, 7	bitrdi 287	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 <s 0s ↔ 0s <s ( -us ‘𝐴)))
9		negscl 28068	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
10		precsex 28242	. . . . . . . 8 ⊢ ((( -us ‘𝐴) ∈ No ∧ 0s <s ( -us ‘𝐴)) → ∃𝑦 ∈ No (( -us ‘𝐴) ·s 𝑦) = 1s )
11	9, 10	sylan 580	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) → ∃𝑦 ∈ No (( -us ‘𝐴) ·s 𝑦) = 1s )
12		simprl 771	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -us ‘𝐴) ·s 𝑦) = 1s )) → 𝑦 ∈ No )
13	12	negscld 28069	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -us ‘𝐴) ·s 𝑦) = 1s )) → ( -us ‘𝑦) ∈ No )
14		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → 𝐴 ∈ No )
15		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → 𝑦 ∈ No )
16	14, 15	mulnegs1d 28186	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (( -us ‘𝐴) ·s 𝑦) = ( -us ‘(𝐴 ·s 𝑦)))
17	14, 15	mulnegs2d 28187	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (𝐴 ·s ( -us ‘𝑦)) = ( -us ‘(𝐴 ·s 𝑦)))
18	16, 17	eqtr4d 2780	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (( -us ‘𝐴) ·s 𝑦) = (𝐴 ·s ( -us ‘𝑦)))
19	18	eqeq1d 2739	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → ((( -us ‘𝐴) ·s 𝑦) = 1s ↔ (𝐴 ·s ( -us ‘𝑦)) = 1s ))
20	19	biimpd 229	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → ((( -us ‘𝐴) ·s 𝑦) = 1s → (𝐴 ·s ( -us ‘𝑦)) = 1s ))
21	20	impr 454	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -us ‘𝐴) ·s 𝑦) = 1s )) → (𝐴 ·s ( -us ‘𝑦)) = 1s )
22		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑥 = ( -us ‘𝑦) → (𝐴 ·s 𝑥) = (𝐴 ·s ( -us ‘𝑦)))
23	22	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑥 = ( -us ‘𝑦) → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s ( -us ‘𝑦)) = 1s ))
24	23	rspcev 3622	. . . . . . . 8 ⊢ ((( -us ‘𝑦) ∈ No ∧ (𝐴 ·s ( -us ‘𝑦)) = 1s ) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
25	13, 21, 24	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -us ‘𝐴) ·s 𝑦) = 1s )) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
26	11, 25	rexlimddv 3161	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
27	26	ex 412	. . . . 5 ⊢ (𝐴 ∈ No → ( 0s <s ( -us ‘𝐴) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ))
28	8, 27	sylbid 240	. . . 4 ⊢ (𝐴 ∈ No → (𝐴 <s 0s → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ))
29		precsex 28242	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s <s 𝐴) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
30	29	ex 412	. . . 4 ⊢ (𝐴 ∈ No → ( 0s <s 𝐴 → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ))
31	28, 30	jaod 860	. . 3 ⊢ (𝐴 ∈ No → ((𝐴 <s 0s ∨ 0s <s 𝐴) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ))
32	3, 31	sylbid 240	. 2 ⊢ (𝐴 ∈ No → (𝐴 ≠ 0s → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ))
33	32	imp 406	1 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431   No csur 27684   <s cslt 27685   0_s c0s 27867   1_s c1s 27868   -u_s cnegs 28051   ·_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  ax-dc 10486

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-ot 4635  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-lim 6389  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-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133  df-divs 28214

This theorem is referenced by:  recsexd  28244  divsmul  28245  divscl  28247