Theorem recsex 28160

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 27788	. . . 4 ⊢ 0s ∈ No
2		slttrine 27713	. . . 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 27994	. . . . . . 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 27975	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
7	6	breq1i 5126	. . . . . 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 27985	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
10		precsex 28159	. . . . . . . 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 27986	. . . . . . . 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 28103	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (( -us ‘𝐴) ·s 𝑦) = ( -us ‘(𝐴 ·s 𝑦)))
17	14, 15	mulnegs2d 28104	. . . . . . . . . . . 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 7411	. . . . . . . . . 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 3601	. . . . . . . 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 3147	. . . . . 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 28159	. . . . 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 2108   ≠ wne 2932  ∃wrex 3060   class class class wbr 5119  ‘cfv 6530  (class class class)co 7403   No csur 27601   <s cslt 27602   0_s c0s 27784   1_s c1s 27785   -u_s cnegs 27968   ·_s cmuls 28049

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-dc 10458

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-oadd 8482  df-nadd 8676  df-no 27604  df-slt 27605  df-bday 27606  df-sle 27707  df-sslt 27743  df-scut 27745  df-0s 27786  df-1s 27787  df-made 27803  df-old 27804  df-left 27806  df-right 27807  df-norec 27888  df-norec2 27899  df-adds 27910  df-negs 27970  df-subs 27971  df-muls 28050  df-divs 28131

This theorem is referenced by:  recsexd  28161  divsmul  28162  divscl  28164