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Theorem expsne0 28429
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 2737 . . . . . . . 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 2737 . . . . . . . 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 2737 . . . . . . . 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 2737 . . . . . . . 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 27889 . . . . . . . . . 10 0s <s 1s
18 sgt0ne0 27894 . . . . . . . . . 10 ( 0s <s 1s → 1s ≠ 0s )
1917, 18ax-mp 5 . . . . . . . . 9 1s ≠ 0s
20 exps0 28425 . . . . . . . . . 10 (𝐴 No → (𝐴s 0s ) = 1s )
2120neeq1d 2998 . . . . . . . . 9 (𝐴 No → ((𝐴s 0s ) ≠ 0s ↔ 1s ≠ 0s ))
2219, 21mpbiri 258 . . . . . . . 8 (𝐴 No → (𝐴s 0s ) ≠ 0s )
2322neneqd 2943 . . . . . . 7 (𝐴 No → ¬ (𝐴s 0s ) = 0s )
2423pm2.21d 121 . . . . . 6 (𝐴 No → ((𝐴s 0s ) = 0s𝐴 = 0s ))
25 expsp1 28427 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕ0s) → (𝐴s(𝑛 +s 1s )) = ((𝐴s𝑛) ·s 𝐴))
2625eqeq1d 2737 . . . . . . . . . . . 12 ((𝐴 No 𝑛 ∈ ℕ0s) → ((𝐴s(𝑛 +s 1s )) = 0s ↔ ((𝐴s𝑛) ·s 𝐴) = 0s ))
27 expscl 28428 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕ0s) → (𝐴s𝑛) ∈ No )
28 simpl 482 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕ0s) → 𝐴 No )
2927, 28muls0ord 28226 . . . . . . . . . . . 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 28352 . . . . 5 (𝑁 ∈ ℕ0s → (𝐴 No → ((𝐴s𝑁) = 0s𝐴 = 0s )))
4039imp 406 . . . 4 ((𝑁 ∈ ℕ0s𝐴 No ) → ((𝐴s𝑁) = 0s𝐴 = 0s ))
4140necon3d 2959 . . 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 1537  wcel 2106  wne 2938   class class class wbr 5148  (class class class)co 7431   No csur 27699   <s cslt 27700   0s c0s 27882   1s c1s 27883   +s cadds 28007   ·s cmuls 28147  0scnn0s 28333  scexps 28411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148  df-seqs 28305  df-n0s 28335  df-nns 28336  df-zs 28380  df-exps 28412
This theorem is referenced by:  cutpw2  28432  pw2bday  28433  pw2cut  28435  zs12bday  28439
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