Theorem recsne0 28608

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 7451	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ·s 𝑥) = (𝐴 ·s 𝑦))
3	2	eqeq1d 2765	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s 𝑦) = 1s ))
4	3	cbvrexvw 3245	. . 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 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → (𝐴 ·s 𝑦) = 1s )
7		1ne0s 28236	. . . . . 6 ⊢ 1s ≠ 0s
8	7	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 1s ≠ 0s )
9	6, 8	eqnetrd 3026	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → (𝐴 ·s 𝑦) ≠ 0s )
10		recsne0.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No)
11	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝐴 ∈ No)
12		simprl 783	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝑦 ∈ No)
13	11, 12	mulsne0bd 28602	. . . 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 500	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝐴 ≠ 0s )
16	5, 15	rexlimddv 3173	1 ⊢ (𝜑 → 𝐴 ≠ 0s )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090  (class class class)co 7443  Nocsur 28027   0_s c0s 28221   1_s c1s 28222   ·_s cmuls 28522

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-subs 28438  df-muls 28523

This theorem is used by: (None)