Theorem norecdiv 28576

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 28521	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → (𝑤 ·s 𝐵) ∈ No)
4		oveq1 7427	. . . . . . . 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 28553	. . . . . 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 28529	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → ( 1s ·s 𝐵) = 𝐵)
10	6, 8, 9	3eqtr3d 2804	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵)
11		oveq2 7428	. . . . . . 7 ⊢ (𝑧 = (𝑤 ·s 𝐵) → (𝐴 ·s 𝑧) = (𝐴 ·s (𝑤 ·s 𝐵)))
12	11	eqeq1d 2763	. . . . . 6 ⊢ (𝑧 = (𝑤 ·s 𝐵) → ((𝐴 ·s 𝑧) = 𝐵 ↔ (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵))
13	12	rspcev 3577	. . . . 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 3165	. . 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 7428	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝐴 ·s 𝑥) = (𝐴 ·s 𝑤))
18	17	eqeq1d 2763	. . . 4 ⊢ (𝑥 = 𝑤 → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s 𝑤) = 1s ))
19	18	cbvrexvw 3242	. . 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 7428	. . . 4 ⊢ (𝑦 = 𝑧 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝑧))
22	21	eqeq1d 2763	. . 3 ⊢ (𝑦 = 𝑧 → ((𝐴 ·s 𝑦) = 𝐵 ↔ (𝐴 ·s 𝑧) = 𝐵))
23	22	cbvrexvw 3242	. 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 2956  ∃wrex 3087  (class class class)co 7420  Nocsur 27997   0_s c0s 28191   1_s c1s 28192   ·_s cmuls 28492

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  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-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  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

This theorem is used by:  noreceuw  28577