Theorem recsex 28128

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 27741	. . . 4 ⊢ 0s ∈ No
2		slttrine 27661	. . . 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 27958	. . . . . . 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 27939	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
7	6	breq1i 5099	. . . . . 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 27949	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
10		precsex 28127	. . . . . . . 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 770	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -us ‘𝐴) ·s 𝑦) = 1s )) → 𝑦 ∈ No )
13	12	negscld 27950	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -us ‘𝐴) ·s 𝑦) = 1s )) → ( -us ‘𝑦) ∈ No )
14		simpll 766	. . . . . . . . . . . . 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 28070	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (( -us ‘𝐴) ·s 𝑦) = ( -us ‘(𝐴 ·s 𝑦)))
17	14, 15	mulnegs2d 28071	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (𝐴 ·s ( -us ‘𝑦)) = ( -us ‘(𝐴 ·s 𝑦)))
18	16, 17	eqtr4d 2767	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (( -us ‘𝐴) ·s 𝑦) = (𝐴 ·s ( -us ‘𝑦)))
19	18	eqeq1d 2731	. . . . . . . . . 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 7357	. . . . . . . . . 10 ⊢ (𝑥 = ( -us ‘𝑦) → (𝐴 ·s 𝑥) = (𝐴 ·s ( -us ‘𝑦)))
23	22	eqeq1d 2731	. . . . . . . . 9 ⊢ (𝑥 = ( -us ‘𝑦) → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s ( -us ‘𝑦)) = 1s ))
24	23	rspcev 3577	. . . . . . . 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 3136	. . . . . 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 28127	. . . . 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 859	. . 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 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349   No csur 27549   <s cslt 27550   0_s c0s 27737   1_s c1s 27738   -u_s cnegs 27932   ·_s cmuls 28016

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-dc 10340

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-nadd 8584  df-no 27552  df-slt 27553  df-bday 27554  df-sle 27655  df-sslt 27692  df-scut 27694  df-0s 27739  df-1s 27740  df-made 27759  df-old 27760  df-left 27762  df-right 27763  df-norec 27852  df-norec2 27863  df-adds 27874  df-negs 27934  df-subs 27935  df-muls 28017  df-divs 28098

This theorem is referenced by:  recsexd  28129  divsmul  28130  divscl  28132