Theorem recsne0 28396

Description: If a surreal has a reciprocal, then it is nonzero. (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 7420	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ·s 𝑥) = (𝐴 ·s 𝑦))
3	2	eqeq1d 2764	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s 𝑦) = 1s ))
4	3	cbvrexvw 3243	. . 3 ⊢ (∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ↔ ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 1s )
5	1, 4	sylib 221	. 2 ⊢ (𝜑 → ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 1s )
6		simprr 784	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → (𝐴 ·s 𝑦) = 1s )
7		1ne0s 28024	. . . . . 6 ⊢ 1s ≠ 0s
8	7	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 1s ≠ 0s )
9	6, 8	eqnetrd 3024	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → (𝐴 ·s 𝑦) ≠ 0s )
10		recsne0.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
11	10	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝐴 ∈ No )
12		simprl 782	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝑦 ∈ No )
13	11, 12	mulsne0bd 28390	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → ((𝐴 ·s 𝑦) ≠ 0s ↔ (𝐴 ≠ 0s ∧ 𝑦 ≠ 0s )))
14	9, 13	mpbid 235	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → (𝐴 ≠ 0s ∧ 𝑦 ≠ 0s ))
15	14	simpld 499	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝐴 ≠ 0s )
16	5, 15	rexlimddv 3171	1 ⊢ (𝜑 → 𝐴 ≠ 0s )

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3088  (class class class)co 7412   No csur 27815   0_s c0s 28009   1_s c1s 28010   ·_s cmuls 28310

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-1o 8451  df-2o 8452  df-nadd 8650  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-0s 28011  df-1s 28012  df-made 28031  df-old 28032  df-left 28034  df-right 28035  df-norec 28142  df-norec2 28153  df-adds 28164  df-negs 28225  df-subs 28226  df-muls 28311

This theorem is used by: (None)