Theorem norecdiv 28458

Description: If a surreal has a reciprocal, then it has any division. (Contributed by Scott Fenton, 12-Mar-2025.)

Assertion

Ref	Expression
norecdiv	⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ) → ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝐵

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem norecdiv

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 783	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → 𝑤 ∈ No )
2		simpl3 1212	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → 𝐵 ∈ No )
3	1, 2	mulscld 28403	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → (𝑤 ·s 𝐵) ∈ No )
4		oveq1 7421	. . . . . . . 8 ⊢ ((𝐴 ·s 𝑤) = 1s → ((𝐴 ·s 𝑤) ·s 𝐵) = ( 1s ·s 𝐵))
5	4	adantl 487	. . . . . . 7 ⊢ ((𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s ) → ((𝐴 ·s 𝑤) ·s 𝐵) = ( 1s ·s 𝐵))
6	5	adantl 487	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → ((𝐴 ·s 𝑤) ·s 𝐵) = ( 1s ·s 𝐵))
7		simpl1 1210	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → 𝐴 ∈ No )
8	7, 1, 2	mulsassd 28435	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → ((𝐴 ·s 𝑤) ·s 𝐵) = (𝐴 ·s (𝑤 ·s 𝐵)))
9	2	mulslidd 28411	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → ( 1s ·s 𝐵) = 𝐵)
10	6, 8, 9	3eqtr3d 2803	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵)
11		oveq2 7422	. . . . . . 7 ⊢ (𝑧 = (𝑤 ·s 𝐵) → (𝐴 ·s 𝑧) = (𝐴 ·s (𝑤 ·s 𝐵)))
12	11	eqeq1d 2762	. . . . . 6 ⊢ (𝑧 = (𝑤 ·s 𝐵) → ((𝐴 ·s 𝑧) = 𝐵 ↔ (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵))
13	12	rspcev 3576	. . . . 5 ⊢ (((𝑤 ·s 𝐵) ∈ No ∧ (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵) → ∃𝑧 ∈ No (𝐴 ·s 𝑧) = 𝐵)
14	3, 10, 13	syl2anc 596	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → ∃𝑧 ∈ No (𝐴 ·s 𝑧) = 𝐵)
15	14	rexlimdvaa 3164	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) → (∃𝑤 ∈ No (𝐴 ·s 𝑤) = 1s → ∃𝑧 ∈ No (𝐴 ·s 𝑧) = 𝐵))
16	15	imp 412	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃𝑤 ∈ No (𝐴 ·s 𝑤) = 1s ) → ∃𝑧 ∈ No (𝐴 ·s 𝑧) = 𝐵)
17		oveq2 7422	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝐴 ·s 𝑥) = (𝐴 ·s 𝑤))
18	17	eqeq1d 2762	. . . 4 ⊢ (𝑥 = 𝑤 → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s 𝑤) = 1s ))
19	18	cbvrexvw 3241	. . 3 ⊢ (∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ↔ ∃𝑤 ∈ No (𝐴 ·s 𝑤) = 1s )
20	19	anbi2i 635	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ) ↔ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃𝑤 ∈ No (𝐴 ·s 𝑤) = 1s ))
21		oveq2 7422	. . . 4 ⊢ (𝑦 = 𝑧 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝑧))
22	21	eqeq1d 2762	. . 3 ⊢ (𝑦 = 𝑧 → ((𝐴 ·s 𝑦) = 𝐵 ↔ (𝐴 ·s 𝑧) = 𝐵))
23	22	cbvrexvw 3241	. 2 ⊢ (∃𝑦 ∈ No (𝐴 ·s 𝑦) = 𝐵 ↔ ∃𝑧 ∈ No (𝐴 ·s 𝑧) = 𝐵)
24	16, 20, 23	3imtr4i 295	1 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ) → ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086  (class class class)co 7414   No csur 27879   0_s c0s 28073   1_s c1s 28074   ·_s cmuls 28374

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

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 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6299  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-1o 8458  df-2o 8459  df-nadd 8657  df-no 27882  df-lts 27883  df-bday 27884  df-les 27984  df-slts 28026  df-cuts 28028  df-0s 28075  df-1s 28076  df-made 28095  df-old 28096  df-left 28098  df-right 28099  df-norec 28206  df-norec2 28217  df-adds 28228  df-negs 28289  df-subs 28290  df-muls 28375

This theorem is used by:  noreceuw  28459