Theorem recsne0 28184

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 7375	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴 ·s 𝑥) = (𝐴 ·s 𝑦))
3	2	eqeq1d 2739	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐴 ·s 𝑥) = 1s ↔ (𝐴 ·s 𝑦) = 1s ))
4	3	cbvrexvw 3217	. . 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 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → (𝐴 ·s 𝑦) = 1s )
7		1ne0s 27812	. . . . . 6 ⊢ 1s ≠ 0s
8	7	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 1s ≠ 0s )
9	6, 8	eqnetrd 3000	. . . 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 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ No ∧ (𝐴 ·s 𝑦) = 1s )) → 𝑦 ∈ No )
13	11, 12	mulsne0bd 28178	. . . 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 3145	1 ⊢ (𝜑 → 𝐴 ≠ 0s )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  (class class class)co 7367   No csur 27603   0_s c0s 27797   1_s c1s 27798   ·_s cmuls 28098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  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 6266  df-ord 6327  df-on 6328  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-nadd 8602  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-1s 27800  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-subs 28014  df-muls 28099

This theorem is referenced by: (None)