Theorem norecdiv 28283

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 780	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → 𝑤 ∈ No )
2		simpl3 1207	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → 𝐵 ∈ No )
3	1, 2	mulscld 28228	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → (𝑤 ·s 𝐵) ∈ No )
4		oveq1 7403	. . . . . . . 8 ⊢ ((𝐴 ·s 𝑤) = 1s → ((𝐴 ·s 𝑤) ·s 𝐵) = ( 1s ·s 𝐵))
5	4	adantl 485	. . . . . . 7 ⊢ ((𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s ) → ((𝐴 ·s 𝑤) ·s 𝐵) = ( 1s ·s 𝐵))
6	5	adantl 485	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → ((𝐴 ·s 𝑤) ·s 𝐵) = ( 1s ·s 𝐵))
7		simpl1 1205	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → 𝐴 ∈ No )
8	7, 1, 2	mulsassd 28260	. . . . . 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 28236	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → ( 1s ·s 𝐵) = 𝐵)
10	6, 8, 9	3eqtr3d 2805	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵)
11		oveq2 7404	. . . . . . 7 ⊢ (𝑧 = (𝑤 ·s 𝐵) → (𝐴 ·s 𝑧) = (𝐴 ·s (𝑤 ·s 𝐵)))
12	11	eqeq1d 2764	. . . . . 6 ⊢ (𝑧 = (𝑤 ·s 𝐵) → ((𝐴 ·s 𝑧) = 𝐵 ↔ (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵))
13	12	rspcev 3581	. . . . 5 ⊢ (((𝑤 ·s 𝐵) ∈ No ∧ (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵) → ∃𝑧 ∈ No (𝐴 ·s 𝑧) = 𝐵)
14	3, 10, 13	syl2anc 593	. . . 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 410	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃𝑤 ∈ No (𝐴 ·s 𝑤) = 1s ) → ∃𝑧 ∈ No (𝐴 ·s 𝑧) = 𝐵)
17		oveq2 7404	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝐴 ·s 𝑥) = (𝐴 ·s 𝑤))
18	17	eqeq1d 2764	. . . 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 632	. 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 7404	. . . 4 ⊢ (𝑦 = 𝑧 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝑧))
22	21	eqeq1d 2764	. . 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 294	1 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ) → ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3086  (class class class)co 7396   No csur 27704   0_s c0s 27898   1_s c1s 27899   ·_s cmuls 28199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-1o 8437  df-2o 8438  df-nadd 8636  df-no 27707  df-lts 27708  df-bday 27709  df-les 27809  df-slts 27851  df-cuts 27853  df-0s 27900  df-1s 27901  df-made 27920  df-old 27921  df-left 27923  df-right 27924  df-norec 28031  df-norec2 28042  df-adds 28053  df-negs 28114  df-subs 28115  df-muls 28200

This theorem is referenced by:  noreceuw  28284