Theorem norecdiv 28633

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 28578	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → (𝑤 ·s 𝐵) ∈ No)
4		oveq1 7458	. . . . . . . 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 28610	. . . . . 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 28586	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → ( 1s ·s 𝐵) = 𝐵)
10	6, 8, 9	3eqtr3d 2806	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵)
11		oveq2 7459	. . . . . . 7 ⊢ (𝑧 = (𝑤 ·s 𝐵) → (𝐴 ·s 𝑧) = (𝐴 ·s (𝑤 ·s 𝐵)))
12	11	eqeq1d 2765	. . . . . 6 ⊢ (𝑧 = (𝑤 ·s 𝐵) → ((𝐴 ·s 𝑧) = 𝐵 ↔ (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵))
13	12	rspcev 3580	. . . . 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 3168	. . 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 7459	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝐴 ·s 𝑥) = (𝐴 ·s 𝑤))
18	17	eqeq1d 2765	. . . 4 ⊢ (𝑥 = 𝑤 → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s 𝑤) = 1s ))
19	18	cbvrexvw 3245	. . 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 7459	. . . 4 ⊢ (𝑦 = 𝑧 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝑧))
22	21	eqeq1d 2765	. . 3 ⊢ (𝑦 = 𝑧 → ((𝐴 ·s 𝑦) = 𝐵 ↔ (𝐴 ·s 𝑧) = 𝐵))
23	22	cbvrexvw 3245	. 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 2147   ≠ wne 2959  ∃wrex 3090  (class class class)co 7451  Nocsur 28054   0_s c0s 28248   1_s c1s 28249   ·_s cmuls 28549

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-nadd 8713  df-no 28057  df-lts 28058  df-bday 28059  df-les 28159  df-slts 28201  df-cuts 28203  df-0s 28250  df-1s 28251  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec 28381  df-norec2 28392  df-adds 28403  df-negs 28464  df-subs 28465  df-muls 28550

This theorem is used by:  noreceuw  28634