Theorem recsex 28198

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 27805	. . . 4 ⊢ 0s ∈ No
2		slttrine 27721	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≠ 0s ↔ (𝐴 <s 0s ∨ 0s <s 𝐴)))
3	1, 2	mpan2 692	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ≠ 0s ↔ (𝐴 <s 0s ∨ 0s <s 𝐴)))
4		sltneg 28025	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐴)))
5	1, 4	mpan2 692	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐴)))
6		negs0s 28006	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
7	6	breq1i 5104	. . . . . 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 28016	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
10		precsex 28197	. . . . . . . 8 ⊢ ((( -us ‘𝐴) ∈ No ∧ 0s <s ( -us ‘𝐴)) → ∃𝑦 ∈ No (( -us ‘𝐴) ·s 𝑦) = 1s )
11	9, 10	sylan 581	. . . . . . 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 28017	. . . . . . . 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 28140	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (( -us ‘𝐴) ·s 𝑦) = ( -us ‘(𝐴 ·s 𝑦)))
17	14, 15	mulnegs2d 28141	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (𝐴 ·s ( -us ‘𝑦)) = ( -us ‘(𝐴 ·s 𝑦)))
18	16, 17	eqtr4d 2773	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (( -us ‘𝐴) ·s 𝑦) = (𝐴 ·s ( -us ‘𝑦)))
19	18	eqeq1d 2737	. . . . . . . . . 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 7366	. . . . . . . . . 10 ⊢ (𝑥 = ( -us ‘𝑦) → (𝐴 ·s 𝑥) = (𝐴 ·s ( -us ‘𝑦)))
23	22	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑥 = ( -us ‘𝑦) → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s ( -us ‘𝑦)) = 1s ))
24	23	rspcev 3575	. . . . . . . 8 ⊢ ((( -us ‘𝑦) ∈ No ∧ (𝐴 ·s ( -us ‘𝑦)) = 1s ) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
25	13, 21, 24	syl2anc 585	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -us ‘𝐴) ·s 𝑦) = 1s )) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
26	11, 25	rexlimddv 3142	. . . . . 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 28197	. . . . 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 1542   ∈ wcel 2114   ≠ wne 2931  ∃wrex 3059   class class class wbr 5097  ‘cfv 6491  (class class class)co 7358   No csur 27609   <s cslt 27610   0_s c0s 27801   1_s c1s 27802   -u_s cnegs 27999   ·_s cmuls 28086

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-dc 10358

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-nadd 8594  df-no 27612  df-slt 27613  df-bday 27614  df-sle 27715  df-sslt 27756  df-scut 27758  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27918  df-norec2 27929  df-adds 27940  df-negs 28001  df-subs 28002  df-muls 28087  df-divs 28168

This theorem is referenced by:  recsexd  28199  divsmul  28200  divscl  28202