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Theorem expsne0 28414
Description: A non-negative surreal integer power is non-zero if its base is non-zero. (Contributed by Scott Fenton, 7-Aug-2025.)
Assertion
Ref Expression
expsne0 ((𝐴 No 𝐴 ≠ 0s𝑁 ∈ ℕ0s) → (𝐴s𝑁) ≠ 0s )

Proof of Theorem expsne0
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7439 . . . . . . . . 9 (𝑚 = 0s → (𝐴s𝑚) = (𝐴s 0s ))
21eqeq1d 2739 . . . . . . . 8 (𝑚 = 0s → ((𝐴s𝑚) = 0s ↔ (𝐴s 0s ) = 0s ))
32imbi1d 341 . . . . . . 7 (𝑚 = 0s → (((𝐴s𝑚) = 0s𝐴 = 0s ) ↔ ((𝐴s 0s ) = 0s𝐴 = 0s )))
43imbi2d 340 . . . . . 6 (𝑚 = 0s → ((𝐴 No → ((𝐴s𝑚) = 0s𝐴 = 0s )) ↔ (𝐴 No → ((𝐴s 0s ) = 0s𝐴 = 0s ))))
5 oveq2 7439 . . . . . . . . 9 (𝑚 = 𝑛 → (𝐴s𝑚) = (𝐴s𝑛))
65eqeq1d 2739 . . . . . . . 8 (𝑚 = 𝑛 → ((𝐴s𝑚) = 0s ↔ (𝐴s𝑛) = 0s ))
76imbi1d 341 . . . . . . 7 (𝑚 = 𝑛 → (((𝐴s𝑚) = 0s𝐴 = 0s ) ↔ ((𝐴s𝑛) = 0s𝐴 = 0s )))
87imbi2d 340 . . . . . 6 (𝑚 = 𝑛 → ((𝐴 No → ((𝐴s𝑚) = 0s𝐴 = 0s )) ↔ (𝐴 No → ((𝐴s𝑛) = 0s𝐴 = 0s ))))
9 oveq2 7439 . . . . . . . . 9 (𝑚 = (𝑛 +s 1s ) → (𝐴s𝑚) = (𝐴s(𝑛 +s 1s )))
109eqeq1d 2739 . . . . . . . 8 (𝑚 = (𝑛 +s 1s ) → ((𝐴s𝑚) = 0s ↔ (𝐴s(𝑛 +s 1s )) = 0s ))
1110imbi1d 341 . . . . . . 7 (𝑚 = (𝑛 +s 1s ) → (((𝐴s𝑚) = 0s𝐴 = 0s ) ↔ ((𝐴s(𝑛 +s 1s )) = 0s𝐴 = 0s )))
1211imbi2d 340 . . . . . 6 (𝑚 = (𝑛 +s 1s ) → ((𝐴 No → ((𝐴s𝑚) = 0s𝐴 = 0s )) ↔ (𝐴 No → ((𝐴s(𝑛 +s 1s )) = 0s𝐴 = 0s ))))
13 oveq2 7439 . . . . . . . . 9 (𝑚 = 𝑁 → (𝐴s𝑚) = (𝐴s𝑁))
1413eqeq1d 2739 . . . . . . . 8 (𝑚 = 𝑁 → ((𝐴s𝑚) = 0s ↔ (𝐴s𝑁) = 0s ))
1514imbi1d 341 . . . . . . 7 (𝑚 = 𝑁 → (((𝐴s𝑚) = 0s𝐴 = 0s ) ↔ ((𝐴s𝑁) = 0s𝐴 = 0s )))
1615imbi2d 340 . . . . . 6 (𝑚 = 𝑁 → ((𝐴 No → ((𝐴s𝑚) = 0s𝐴 = 0s )) ↔ (𝐴 No → ((𝐴s𝑁) = 0s𝐴 = 0s ))))
17 0slt1s 27874 . . . . . . . . . 10 0s <s 1s
18 sgt0ne0 27879 . . . . . . . . . 10 ( 0s <s 1s → 1s ≠ 0s )
1917, 18ax-mp 5 . . . . . . . . 9 1s ≠ 0s
20 exps0 28410 . . . . . . . . . 10 (𝐴 No → (𝐴s 0s ) = 1s )
2120neeq1d 3000 . . . . . . . . 9 (𝐴 No → ((𝐴s 0s ) ≠ 0s ↔ 1s ≠ 0s ))
2219, 21mpbiri 258 . . . . . . . 8 (𝐴 No → (𝐴s 0s ) ≠ 0s )
2322neneqd 2945 . . . . . . 7 (𝐴 No → ¬ (𝐴s 0s ) = 0s )
2423pm2.21d 121 . . . . . 6 (𝐴 No → ((𝐴s 0s ) = 0s𝐴 = 0s ))
25 expsp1 28412 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕ0s) → (𝐴s(𝑛 +s 1s )) = ((𝐴s𝑛) ·s 𝐴))
2625eqeq1d 2739 . . . . . . . . . . . 12 ((𝐴 No 𝑛 ∈ ℕ0s) → ((𝐴s(𝑛 +s 1s )) = 0s ↔ ((𝐴s𝑛) ·s 𝐴) = 0s ))
27 expscl 28413 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕ0s) → (𝐴s𝑛) ∈ No )
28 simpl 482 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕ0s) → 𝐴 No )
2927, 28muls0ord 28211 . . . . . . . . . . . 12 ((𝐴 No 𝑛 ∈ ℕ0s) → (((𝐴s𝑛) ·s 𝐴) = 0s ↔ ((𝐴s𝑛) = 0s𝐴 = 0s )))
3026, 29bitrd 279 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕ0s) → ((𝐴s(𝑛 +s 1s )) = 0s ↔ ((𝐴s𝑛) = 0s𝐴 = 0s )))
3130adantr 480 . . . . . . . . . 10 (((𝐴 No 𝑛 ∈ ℕ0s) ∧ ((𝐴s𝑛) = 0s𝐴 = 0s )) → ((𝐴s(𝑛 +s 1s )) = 0s ↔ ((𝐴s𝑛) = 0s𝐴 = 0s )))
32 simpr 484 . . . . . . . . . . 11 (((𝐴 No 𝑛 ∈ ℕ0s) ∧ ((𝐴s𝑛) = 0s𝐴 = 0s )) → ((𝐴s𝑛) = 0s𝐴 = 0s ))
33 idd 24 . . . . . . . . . . 11 (((𝐴 No 𝑛 ∈ ℕ0s) ∧ ((𝐴s𝑛) = 0s𝐴 = 0s )) → (𝐴 = 0s𝐴 = 0s ))
3432, 33jaod 860 . . . . . . . . . 10 (((𝐴 No 𝑛 ∈ ℕ0s) ∧ ((𝐴s𝑛) = 0s𝐴 = 0s )) → (((𝐴s𝑛) = 0s𝐴 = 0s ) → 𝐴 = 0s ))
3531, 34sylbid 240 . . . . . . . . 9 (((𝐴 No 𝑛 ∈ ℕ0s) ∧ ((𝐴s𝑛) = 0s𝐴 = 0s )) → ((𝐴s(𝑛 +s 1s )) = 0s𝐴 = 0s ))
3635ex 412 . . . . . . . 8 ((𝐴 No 𝑛 ∈ ℕ0s) → (((𝐴s𝑛) = 0s𝐴 = 0s ) → ((𝐴s(𝑛 +s 1s )) = 0s𝐴 = 0s )))
3736expcom 413 . . . . . . 7 (𝑛 ∈ ℕ0s → (𝐴 No → (((𝐴s𝑛) = 0s𝐴 = 0s ) → ((𝐴s(𝑛 +s 1s )) = 0s𝐴 = 0s ))))
3837a2d 29 . . . . . 6 (𝑛 ∈ ℕ0s → ((𝐴 No → ((𝐴s𝑛) = 0s𝐴 = 0s )) → (𝐴 No → ((𝐴s(𝑛 +s 1s )) = 0s𝐴 = 0s ))))
394, 8, 12, 16, 24, 38n0sind 28337 . . . . 5 (𝑁 ∈ ℕ0s → (𝐴 No → ((𝐴s𝑁) = 0s𝐴 = 0s )))
4039imp 406 . . . 4 ((𝑁 ∈ ℕ0s𝐴 No ) → ((𝐴s𝑁) = 0s𝐴 = 0s ))
4140necon3d 2961 . . 3 ((𝑁 ∈ ℕ0s𝐴 No ) → (𝐴 ≠ 0s → (𝐴s𝑁) ≠ 0s ))
4241ex 412 . 2 (𝑁 ∈ ℕ0s → (𝐴 No → (𝐴 ≠ 0s → (𝐴s𝑁) ≠ 0s )))
43423imp231 1113 1 ((𝐴 No 𝐴 ≠ 0s𝑁 ∈ ℕ0s) → (𝐴s𝑁) ≠ 0s )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  (class class class)co 7431   No csur 27684   <s cslt 27685   0s c0s 27867   1s c1s 27868   +s cadds 27992   ·s cmuls 28132  0scnn0s 28318  scexps 28396
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133  df-seqs 28290  df-n0s 28320  df-nns 28321  df-zs 28365  df-exps 28397
This theorem is referenced by:  cutpw2  28417  pw2bday  28418  pw2cut  28420  zs12bday  28424
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