Theorem recsex 28605

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 28195	. . . 4 ⊢ 0s ∈ No
2		ltstrine 28108	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No) → (𝐴 ≠ 0s ↔ (𝐴 <s 0s ∨ 0s <s 𝐴)))
3	1, 2	mpan2 704	. . 3 ⊢ (𝐴 ∈ No → (𝐴 ≠ 0s ↔ (𝐴 <s 0s ∨ 0s <s 𝐴)))
4		ltnegs 28431	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 0s ∈ No) → (𝐴 <s 0s ↔ ( -s‘ 0s ) <s ( -s‘𝐴)))
5	1, 4	mpan2 704	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴 <s 0s ↔ ( -s‘ 0s ) <s ( -s‘𝐴)))
6		neg0s 28412	. . . . . . 7 ⊢ ( -s‘ 0s ) = 0s
7	6	breq1i 5109	. . . . . 6 ⊢ (( -s‘ 0s ) <s ( -s‘𝐴) ↔ 0s <s ( -s‘𝐴))
8	5, 7	bitrdi 290	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 <s 0s ↔ 0s <s ( -s‘𝐴)))
9		negscl 28422	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( -s‘𝐴) ∈ No)
10		precsex 28604	. . . . . . . 8 ⊢ ((( -s‘𝐴) ∈ No ∧ 0s <s ( -s‘𝐴)) → ∃𝑦 ∈ No (( -s‘𝐴) ·s 𝑦) = 1s )
11	9, 10	sylan 592	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) → ∃𝑦 ∈ No (( -s‘𝐴) ·s 𝑦) = 1s )
12		simprl 783	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -s‘𝐴) ·s 𝑦) = 1s )) → 𝑦 ∈ No)
13	12	negscld 28423	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -s‘𝐴) ·s 𝑦) = 1s )) → ( -s‘𝑦) ∈ No)
14		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ 𝑦 ∈ No) → 𝐴 ∈ No)
15		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ 𝑦 ∈ No) → 𝑦 ∈ No)
16	14, 15	mulnegs1d 28546	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ 𝑦 ∈ No) → (( -s‘𝐴) ·s 𝑦) = ( -s‘(𝐴 ·s 𝑦)))
17	14, 15	mulnegs2d 28547	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ 𝑦 ∈ No) → (𝐴 ·s ( -s‘𝑦)) = ( -s‘(𝐴 ·s 𝑦)))
18	16, 17	eqtr4d 2798	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ 𝑦 ∈ No) → (( -s‘𝐴) ·s 𝑦) = (𝐴 ·s ( -s‘𝑦)))
19	18	eqeq1d 2762	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ 𝑦 ∈ No) → ((( -s‘𝐴) ·s 𝑦) = 1s ↔ (𝐴 ·s ( -s‘𝑦)) = 1s ))
20	19	biimpd 232	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ 𝑦 ∈ No) → ((( -s‘𝐴) ·s 𝑦) = 1s → (𝐴 ·s ( -s‘𝑦)) = 1s ))
21	20	impr 460	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -s‘𝐴) ·s 𝑦) = 1s )) → (𝐴 ·s ( -s‘𝑦)) = 1s )
22		oveq2 7428	. . . . . . . . . 10 ⊢ (𝑥 = ( -s‘𝑦) → (𝐴 ·s 𝑥) = (𝐴 ·s ( -s‘𝑦)))
23	22	eqeq1d 2762	. . . . . . . . 9 ⊢ (𝑥 = ( -s‘𝑦) → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s ( -s‘𝑦)) = 1s ))
24	23	rspcev 3576	. . . . . . . 8 ⊢ ((( -s‘𝑦) ∈ No ∧ (𝐴 ·s ( -s‘𝑦)) = 1s ) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
25	13, 21, 24	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ (𝑦 ∈ No ∧ (( -s‘𝐴) ·s 𝑦) = 1s )) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
26	11, 25	rexlimddv 3169	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
27	26	ex 418	. . . . 5 ⊢ (𝐴 ∈ No → ( 0s <s ( -s‘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 28604	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s <s 𝐴) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
30	29	ex 418	. . . 4 ⊢ (𝐴 ∈ No → ( 0s <s 𝐴 → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ))
31	28, 30	jaod 873	. . 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 412	1 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Nocsur 27997   <s clts 27998   0_s c0s 28191   1_s c1s 28192   -_scnegs 28405   ·_s cmuls 28492

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-dc 10524

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-nadd 8675  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-1s 28194  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-norec2 28335  df-adds 28346  df-negs 28407  df-subs 28408  df-muls 28493  df-divs 28574

This theorem is used by:  recsexd  28606  divmuls  28607  divscl  28609