Theorem recsex 28632

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 28222	. . . 4 ⊢ 0s ∈ No
2		ltstrine 28135	. . . 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 28458	. . . . . . 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 28439	. . . . . . 7 ⊢ ( -s‘ 0s ) = 0s
7	6	breq1i 5113	. . . . . 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 28449	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( -s‘𝐴) ∈ No)
10		precsex 28631	. . . . . . . 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 28450	. . . . . . . 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 28573	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ 𝑦 ∈ No) → (( -s‘𝐴) ·s 𝑦) = ( -s‘(𝐴 ·s 𝑦)))
17	14, 15	mulnegs2d 28574	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ 𝑦 ∈ No) → (𝐴 ·s ( -s‘𝑦)) = ( -s‘(𝐴 ·s 𝑦)))
18	16, 17	eqtr4d 2801	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 0s <s ( -s‘𝐴)) ∧ 𝑦 ∈ No) → (( -s‘𝐴) ·s 𝑦) = (𝐴 ·s ( -s‘𝑦)))
19	18	eqeq1d 2765	. . . . . . . . . 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 7448	. . . . . . . . . 10 ⊢ (𝑥 = ( -s‘𝑦) → (𝐴 ·s 𝑥) = (𝐴 ·s ( -s‘𝑦)))
23	22	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑥 = ( -s‘𝑦) → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s ( -s‘𝑦)) = 1s ))
24	23	rspcev 3580	. . . . . . . 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 3173	. . . . . 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 28631	. . . . 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 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090   class class class wbr 5106  ‘cfv 6555  (class class class)co 7440  Nocsur 28024   <s clts 28025   0_s c0s 28218   1_s c1s 28219   -_scnegs 28432   ·_s cmuls 28519

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-dc 10548

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-nadd 8700  df-no 28027  df-lts 28028  df-bday 28029  df-les 28129  df-slts 28171  df-cuts 28173  df-0s 28220  df-1s 28221  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-norec 28351  df-norec2 28362  df-adds 28373  df-negs 28434  df-subs 28435  df-muls 28520  df-divs 28601

This theorem is used by:  recsexd  28633  divmuls  28634  divscl  28636