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Theorem expsne0 28415
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 7440 . . . . . . . . 9 (𝑚 = 0s → (𝐴s𝑚) = (𝐴s 0s ))
21eqeq1d 2738 . . . . . . . 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 7440 . . . . . . . . 9 (𝑚 = 𝑛 → (𝐴s𝑚) = (𝐴s𝑛))
65eqeq1d 2738 . . . . . . . 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 7440 . . . . . . . . 9 (𝑚 = (𝑛 +s 1s ) → (𝐴s𝑚) = (𝐴s(𝑛 +s 1s )))
109eqeq1d 2738 . . . . . . . 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 7440 . . . . . . . . 9 (𝑚 = 𝑁 → (𝐴s𝑚) = (𝐴s𝑁))
1413eqeq1d 2738 . . . . . . . 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 27875 . . . . . . . . . 10 0s <s 1s
18 sgt0ne0 27880 . . . . . . . . . 10 ( 0s <s 1s → 1s ≠ 0s )
1917, 18ax-mp 5 . . . . . . . . 9 1s ≠ 0s
20 exps0 28411 . . . . . . . . . 10 (𝐴 No → (𝐴s 0s ) = 1s )
2120neeq1d 2999 . . . . . . . . 9 (𝐴 No → ((𝐴s 0s ) ≠ 0s ↔ 1s ≠ 0s ))
2219, 21mpbiri 258 . . . . . . . 8 (𝐴 No → (𝐴s 0s ) ≠ 0s )
2322neneqd 2944 . . . . . . 7 (𝐴 No → ¬ (𝐴s 0s ) = 0s )
2423pm2.21d 121 . . . . . 6 (𝐴 No → ((𝐴s 0s ) = 0s𝐴 = 0s ))
25 expsp1 28413 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕ0s) → (𝐴s(𝑛 +s 1s )) = ((𝐴s𝑛) ·s 𝐴))
2625eqeq1d 2738 . . . . . . . . . . . 12 ((𝐴 No 𝑛 ∈ ℕ0s) → ((𝐴s(𝑛 +s 1s )) = 0s ↔ ((𝐴s𝑛) ·s 𝐴) = 0s ))
27 expscl 28414 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕ0s) → (𝐴s𝑛) ∈ No )
28 simpl 482 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕ0s) → 𝐴 No )
2927, 28muls0ord 28212 . . . . . . . . . . . 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 859 . . . . . . . . . 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 28338 . . . . 5 (𝑁 ∈ ℕ0s → (𝐴 No → ((𝐴s𝑁) = 0s𝐴 = 0s )))
4039imp 406 . . . 4 ((𝑁 ∈ ℕ0s𝐴 No ) → ((𝐴s𝑁) = 0s𝐴 = 0s ))
4140necon3d 2960 . . 3 ((𝑁 ∈ ℕ0s𝐴 No ) → (𝐴 ≠ 0s → (𝐴s𝑁) ≠ 0s ))
4241ex 412 . 2 (𝑁 ∈ ℕ0s → (𝐴 No → (𝐴 ≠ 0s → (𝐴s𝑁) ≠ 0s )))
43423imp231 1112 1 ((𝐴 No 𝐴 ≠ 0s𝑁 ∈ ℕ0s) → (𝐴s𝑁) ≠ 0s )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1539  wcel 2107  wne 2939   class class class wbr 5142  (class class class)co 7432   No csur 27685   <s cslt 27686   0s c0s 27868   1s c1s 27869   +s cadds 27993   ·s cmuls 28133  0scnn0s 28319  scexps 28397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-1s 27871  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-norec2 27983  df-adds 27994  df-negs 28054  df-subs 28055  df-muls 28134  df-seqs 28291  df-n0s 28321  df-nns 28322  df-zs 28366  df-exps 28398
This theorem is referenced by:  cutpw2  28418  pw2bday  28419  pw2cut  28421  zs12bday  28425
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