Theorem norecdiv 28129

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 770	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → 𝑤 ∈ No )
2		simpl3 1194	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → 𝐵 ∈ No )
3	1, 2	mulscld 28074	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → (𝑤 ·s 𝐵) ∈ No )
4		oveq1 7353	. . . . . . . 8 ⊢ ((𝐴 ·s 𝑤) = 1s → ((𝐴 ·s 𝑤) ·s 𝐵) = ( 1s ·s 𝐵))
5	4	adantl 481	. . . . . . 7 ⊢ ((𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s ) → ((𝐴 ·s 𝑤) ·s 𝐵) = ( 1s ·s 𝐵))
6	5	adantl 481	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → ((𝐴 ·s 𝑤) ·s 𝐵) = ( 1s ·s 𝐵))
7		simpl1 1192	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → 𝐴 ∈ No )
8	7, 1, 2	mulsassd 28106	. . . . . 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 28082	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → ( 1s ·s 𝐵) = 𝐵)
10	6, 8, 9	3eqtr3d 2774	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵)
11		oveq2 7354	. . . . . . 7 ⊢ (𝑧 = (𝑤 ·s 𝐵) → (𝐴 ·s 𝑧) = (𝐴 ·s (𝑤 ·s 𝐵)))
12	11	eqeq1d 2733	. . . . . 6 ⊢ (𝑧 = (𝑤 ·s 𝐵) → ((𝐴 ·s 𝑧) = 𝐵 ↔ (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵))
13	12	rspcev 3572	. . . . 5 ⊢ (((𝑤 ·s 𝐵) ∈ No ∧ (𝐴 ·s (𝑤 ·s 𝐵)) = 𝐵) → ∃𝑧 ∈ No (𝐴 ·s 𝑧) = 𝐵)
14	3, 10, 13	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ (𝑤 ∈ No ∧ (𝐴 ·s 𝑤) = 1s )) → ∃𝑧 ∈ No (𝐴 ·s 𝑧) = 𝐵)
15	14	rexlimdvaa 3134	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) → (∃𝑤 ∈ No (𝐴 ·s 𝑤) = 1s → ∃𝑧 ∈ No (𝐴 ·s 𝑧) = 𝐵))
16	15	imp 406	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃𝑤 ∈ No (𝐴 ·s 𝑤) = 1s ) → ∃𝑧 ∈ No (𝐴 ·s 𝑧) = 𝐵)
17		oveq2 7354	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝐴 ·s 𝑥) = (𝐴 ·s 𝑤))
18	17	eqeq1d 2733	. . . 4 ⊢ (𝑥 = 𝑤 → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s 𝑤) = 1s ))
19	18	cbvrexvw 3211	. . 3 ⊢ (∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ↔ ∃𝑤 ∈ No (𝐴 ·s 𝑤) = 1s )
20	19	anbi2i 623	. 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 7354	. . . 4 ⊢ (𝑦 = 𝑧 → (𝐴 ·s 𝑦) = (𝐴 ·s 𝑧))
22	21	eqeq1d 2733	. . 3 ⊢ (𝑦 = 𝑧 → ((𝐴 ·s 𝑦) = 𝐵 ↔ (𝐴 ·s 𝑧) = 𝐵))
23	22	cbvrexvw 3211	. 2 ⊢ (∃𝑦 ∈ No (𝐴 ·s 𝑦) = 𝐵 ↔ ∃𝑧 ∈ No (𝐴 ·s 𝑧) = 𝐵)
24	16, 20, 23	3imtr4i 292	1 ⊢ (((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ) → ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056  (class class class)co 7346   No csur 27578   0_s c0s 27766   1_s c1s 27767   ·_s cmuls 28045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-ot 4582  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-2o 8386  df-nadd 8581  df-no 27581  df-slt 27582  df-bday 27583  df-sle 27684  df-sslt 27721  df-scut 27723  df-0s 27768  df-1s 27769  df-made 27788  df-old 27789  df-left 27791  df-right 27792  df-norec 27881  df-norec2 27892  df-adds 27903  df-negs 27963  df-subs 27964  df-muls 28046

This theorem is referenced by:  noreceuw  28130