Theorem recsex 28423

Description: A nonzero 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		0no 28013	. . . 4 ⊢ 0s ∈ No
2		ltstrine 27926	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 ≠ 0s ↔ (𝐴 <s 0s ∨ 0s <s 𝐴)))
3	1, 2	mpan2 703	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ≠ 0s ↔ (𝐴 <s 0s ∨ 0s <s 𝐴)))
4		ltnegs 28249	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No ) → (𝐴 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐴)))
5	1, 4	mpan2 703	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 <s 0s ↔ ( -us ‘ 0s ) <s ( -us ‘𝐴)))
6		neg0s 28230	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
7	6	breq1i 5115	. . . . . 6 ⊢ (( -us ‘ 0s ) <s ( -us ‘𝐴) ↔ 0s <s ( -us ‘𝐴))
8	5, 7	bitrdi 290	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 <s 0s ↔ 0s <s ( -us ‘𝐴)))
9		negscl 28240	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
10		precsex 28422	. . . . . . . 8 ⊢ ((( -us ‘𝐴) ∈ No ∧ 0s <s ( -us ‘𝐴)) → ∃𝑦 ∈ No (( -us ‘𝐴) ·s 𝑦) = 1s )
11	9, 10	sylan 591	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) → ∃𝑦 ∈ No (( -us ‘𝐴) ·s 𝑦) = 1s )
12		simprl 782	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -us ‘𝐴) ·s 𝑦) = 1s )) → 𝑦 ∈ No )
13	12	negscld 28241	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -us ‘𝐴) ·s 𝑦) = 1s )) → ( -us ‘𝑦) ∈ No )
14		simpll 778	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → 𝐴 ∈ No )
15		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → 𝑦 ∈ No )
16	14, 15	mulnegs1d 28364	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (( -us ‘𝐴) ·s 𝑦) = ( -us ‘(𝐴 ·s 𝑦)))
17	14, 15	mulnegs2d 28365	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (𝐴 ·s ( -us ‘𝑦)) = ( -us ‘(𝐴 ·s 𝑦)))
18	16, 17	eqtr4d 2800	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → (( -us ‘𝐴) ·s 𝑦) = (𝐴 ·s ( -us ‘𝑦)))
19	18	eqeq1d 2764	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → ((( -us ‘𝐴) ·s 𝑦) = 1s ↔ (𝐴 ·s ( -us ‘𝑦)) = 1s ))
20	19	biimpd 232	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ 𝑦 ∈ No ) → ((( -us ‘𝐴) ·s 𝑦) = 1s → (𝐴 ·s ( -us ‘𝑦)) = 1s ))
21	20	impr 459	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -us ‘𝐴) ·s 𝑦) = 1s )) → (𝐴 ·s ( -us ‘𝑦)) = 1s )
22		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑥 = ( -us ‘𝑦) → (𝐴 ·s 𝑥) = (𝐴 ·s ( -us ‘𝑦)))
23	22	eqeq1d 2764	. . . . . . . . 9 ⊢ (𝑥 = ( -us ‘𝑦) → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s ( -us ‘𝑦)) = 1s ))
24	23	rspcev 3580	. . . . . . . 8 ⊢ ((( -us ‘𝑦) ∈ No ∧ (𝐴 ·s ( -us ‘𝑦)) = 1s ) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
25	13, 21, 24	syl2anc 595	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -us ‘𝐴) ·s 𝑦) = 1s )) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
26	11, 25	rexlimddv 3171	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 0s <s ( -us ‘𝐴)) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
27	26	ex 417	. . . . 5 ⊢ (𝐴 ∈ No → ( 0s <s ( -us ‘𝐴) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ))
28	8, 27	sylbid 243	. . . 4 ⊢ (𝐴 ∈ No → (𝐴 <s 0s → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ))
29		precsex 28422	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s <s 𝐴) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
30	29	ex 417	. . . 4 ⊢ (𝐴 ∈ No → ( 0s <s 𝐴 → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ))
31	28, 30	jaod 872	. . 3 ⊢ (𝐴 ∈ No → ((𝐴 <s 0s ∨ 0s <s 𝐴) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ))
32	3, 31	sylbid 243	. 2 ⊢ (𝐴 ∈ No → (𝐴 ≠ 0s → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ))
33	32	imp 411	1 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3088   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412   No csur 27815   <s clts 27816   0_s c0s 28009   1_s c1s 28010   -u_s cnegs 28223   ·_s cmuls 28310

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-dc 10436

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-oadd 8455  df-nadd 8650  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-0s 28011  df-1s 28012  df-made 28031  df-old 28032  df-left 28034  df-right 28035  df-norec 28142  df-norec2 28153  df-adds 28164  df-negs 28225  df-subs 28226  df-muls 28311  df-divs 28392

This theorem is used by:  recsexd  28424  divmuls  28425  divscl  28427