Theorem recsne0 28251

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 7389	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ·s 𝑥) = (𝐴 ·s 𝑦))
3	2	eqeq1d 2754	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s 𝑦) = 1s ))
4	3	cbvrexvw 3231	. . 3 ⊢ (∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ↔ ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 1s )
5	1, 4	sylib 220	. 2 ⊢ (𝜑 → ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 1s )
6		simprr 780	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → (𝐴 ·s 𝑦) = 1s )
7		1ne0s 27879	. . . . . 6 ⊢ 1s ≠ 0s
8	7	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 1s ≠ 0s )
9	6, 8	eqnetrd 3014	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → (𝐴 ·s 𝑦) ≠ 0s )
10		recsne0.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
11	10	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝐴 ∈ No )
12		simprl 778	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝑦 ∈ No )
13	11, 12	mulsne0bd 28245	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → ((𝐴 ·s 𝑦) ≠ 0s ↔ (𝐴 ≠ 0s ∧ 𝑦 ≠ 0s )))
14	9, 13	mpbid 234	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → (𝐴 ≠ 0s ∧ 𝑦 ≠ 0s ))
15	14	simpld 497	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝐴 ≠ 0s )
16	5, 15	rexlimddv 3159	1 ⊢ (𝜑 → 𝐴 ≠ 0s )

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∃wrex 3076  (class class class)co 7381   No csur 27670   0_s c0s 27864   1_s c1s 27865   ·_s cmuls 28165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-ot 4581  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-1o 8421  df-2o 8422  df-nadd 8620  df-no 27673  df-lts 27674  df-bday 27675  df-les 27775  df-slts 27817  df-cuts 27819  df-0s 27866  df-1s 27867  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27997  df-norec2 28008  df-adds 28019  df-negs 28080  df-subs 28081  df-muls 28166

This theorem is referenced by: (None)