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Theorem algcvgb 11803
 Description: Two ways of expressing that 𝐶 is a countdown function for algorithm 𝐹. The first is used in these theorems. The second states the condition more intuitively as a conjunction: if the countdown function's value is currently nonzero, it must decrease at the next step; if it has reached zero, it must remain zero at the next step. (Contributed by Paul Chapman, 31-Mar-2011.)
Hypotheses
Ref Expression
algcvgb.1 𝐹:𝑆𝑆
algcvgb.2 𝐶:𝑆⟶ℕ0
Assertion
Ref Expression
algcvgb (𝑋𝑆 → (((𝐶‘(𝐹𝑋)) ≠ 0 → (𝐶‘(𝐹𝑋)) < (𝐶𝑋)) ↔ (((𝐶𝑋) ≠ 0 → (𝐶‘(𝐹𝑋)) < (𝐶𝑋)) ∧ ((𝐶𝑋) = 0 → (𝐶‘(𝐹𝑋)) = 0))))

Proof of Theorem algcvgb
StepHypRef Expression
1 algcvgb.2 . . 3 𝐶:𝑆⟶ℕ0
21ffvelrni 5566 . 2 (𝑋𝑆 → (𝐶𝑋) ∈ ℕ0)
3 algcvgb.1 . . . 4 𝐹:𝑆𝑆
43ffvelrni 5566 . . 3 (𝑋𝑆 → (𝐹𝑋) ∈ 𝑆)
51ffvelrni 5566 . . 3 ((𝐹𝑋) ∈ 𝑆 → (𝐶‘(𝐹𝑋)) ∈ ℕ0)
64, 5syl 14 . 2 (𝑋𝑆 → (𝐶‘(𝐹𝑋)) ∈ ℕ0)
7 algcvgblem 11802 . 2 (((𝐶𝑋) ∈ ℕ0 ∧ (𝐶‘(𝐹𝑋)) ∈ ℕ0) → (((𝐶‘(𝐹𝑋)) ≠ 0 → (𝐶‘(𝐹𝑋)) < (𝐶𝑋)) ↔ (((𝐶𝑋) ≠ 0 → (𝐶‘(𝐹𝑋)) < (𝐶𝑋)) ∧ ((𝐶𝑋) = 0 → (𝐶‘(𝐹𝑋)) = 0))))
82, 6, 7syl2anc 409 1 (𝑋𝑆 → (((𝐶‘(𝐹𝑋)) ≠ 0 → (𝐶‘(𝐹𝑋)) < (𝐶𝑋)) ↔ (((𝐶𝑋) ≠ 0 → (𝐶‘(𝐹𝑋)) < (𝐶𝑋)) ∧ ((𝐶𝑋) = 0 → (𝐶‘(𝐹𝑋)) = 0))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 2112   ≠ wne 2310   class class class wbr 3939  ⟶wf 5131  ‘cfv 5135  0cc0 7673   < clt 7853  ℕ0cn0 9030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2114  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142  ax-un 4366  ax-setind 4463  ax-cnex 7764  ax-resscn 7765  ax-1cn 7766  ax-1re 7767  ax-icn 7768  ax-addcl 7769  ax-addrcl 7770  ax-mulcl 7771  ax-addcom 7773  ax-addass 7775  ax-distr 7777  ax-i2m1 7778  ax-0lt1 7779  ax-0id 7781  ax-rnegex 7782  ax-cnre 7784  ax-pre-ltirr 7785  ax-pre-ltwlin 7786  ax-pre-lttrn 7787  ax-pre-apti 7788  ax-pre-ltadd 7789 This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ne 2311  df-nel 2406  df-ral 2423  df-rex 2424  df-reu 2425  df-rab 2427  df-v 2693  df-sbc 2916  df-dif 3080  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-int 3782  df-br 3940  df-opab 4000  df-id 4226  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-iota 5100  df-fun 5137  df-fn 5138  df-f 5139  df-fv 5143  df-riota 5742  df-ov 5789  df-oprab 5790  df-mpo 5791  df-pnf 7855  df-mnf 7856  df-xr 7857  df-ltxr 7858  df-le 7859  df-sub 7988  df-neg 7989  df-inn 8774  df-n0 9031  df-z 9108 This theorem is referenced by:  algcvga  11804
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