Theorem recsne0 28137

Description: If a surreal has a reciprocal, then it is non-zero. (Contributed by Scott Fenton, 5-Sep-2025.)

Hypotheses

Ref	Expression
recsne0.1	⊢ (𝜑 → 𝐴 ∈ No )
recsne0.2	⊢ (𝜑 → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )

Assertion

Ref	Expression
recsne0	⊢ (𝜑 → 𝐴 ≠ 0s )

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem recsne0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		recsne0.2	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
2		oveq2 7378	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ·s 𝑥) = (𝐴 ·s 𝑦))
3	2	eqeq1d 2731	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s 𝑦) = 1s ))
4	3	cbvrexvw 3214	. . 3 ⊢ (∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ↔ ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 1s )
5	1, 4	sylib 218	. 2 ⊢ (𝜑 → ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 1s )
6		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → (𝐴 ·s 𝑦) = 1s )
7		1sne0s 27788	. . . . . 6 ⊢ 1s ≠ 0s
8	7	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 1s ≠ 0s )
9	6, 8	eqnetrd 2992	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → (𝐴 ·s 𝑦) ≠ 0s )
10		recsne0.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
11	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝐴 ∈ No )
12		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝑦 ∈ No )
13	11, 12	mulsne0bd 28131	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → ((𝐴 ·s 𝑦) ≠ 0s ↔ (𝐴 ≠ 0s ∧ 𝑦 ≠ 0s )))
14	9, 13	mpbid 232	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → (𝐴 ≠ 0s ∧ 𝑦 ≠ 0s ))
15	14	simpld 494	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝐴 ≠ 0s )
16	5, 15	rexlimddv 3140	1 ⊢ (𝜑 → 𝐴 ≠ 0s )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  (class class class)co 7370   No csur 27586   0_s c0s 27773   1_s c1s 27774   ·_s cmuls 28051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6263  df-ord 6324  df-on 6325  df-suc 6327  df-iota 6453  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7948  df-2nd 7949  df-frecs 8238  df-wrecs 8269  df-recs 8318  df-1o 8412  df-2o 8413  df-nadd 8608  df-no 27589  df-slt 27590  df-bday 27591  df-sle 27692  df-sslt 27729  df-scut 27731  df-0s 27775  df-1s 27776  df-made 27794  df-old 27795  df-left 27797  df-right 27798  df-norec 27887  df-norec2 27898  df-adds 27909  df-negs 27969  df-subs 27970  df-muls 28052

This theorem is referenced by: (None)