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Theorem algcvgb 12016
Description: Two ways of expressing that 𝐢 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:π‘†βŸΆπ‘†
algcvgb.2 𝐢:π‘†βŸΆβ„•0
Assertion
Ref Expression
algcvgb (𝑋 ∈ 𝑆 β†’ (((πΆβ€˜(πΉβ€˜π‘‹)) β‰  0 β†’ (πΆβ€˜(πΉβ€˜π‘‹)) < (πΆβ€˜π‘‹)) ↔ (((πΆβ€˜π‘‹) β‰  0 β†’ (πΆβ€˜(πΉβ€˜π‘‹)) < (πΆβ€˜π‘‹)) ∧ ((πΆβ€˜π‘‹) = 0 β†’ (πΆβ€˜(πΉβ€˜π‘‹)) = 0))))

Proof of Theorem algcvgb
StepHypRef Expression
1 algcvgb.2 . . 3 𝐢:π‘†βŸΆβ„•0
21ffvelcdmi 5642 . 2 (𝑋 ∈ 𝑆 β†’ (πΆβ€˜π‘‹) ∈ β„•0)
3 algcvgb.1 . . . 4 𝐹:π‘†βŸΆπ‘†
43ffvelcdmi 5642 . . 3 (𝑋 ∈ 𝑆 β†’ (πΉβ€˜π‘‹) ∈ 𝑆)
51ffvelcdmi 5642 . . 3 ((πΉβ€˜π‘‹) ∈ 𝑆 β†’ (πΆβ€˜(πΉβ€˜π‘‹)) ∈ β„•0)
64, 5syl 14 . 2 (𝑋 ∈ 𝑆 β†’ (πΆβ€˜(πΉβ€˜π‘‹)) ∈ β„•0)
7 algcvgblem 12015 . 2 (((πΆβ€˜π‘‹) ∈ β„•0 ∧ (πΆβ€˜(πΉβ€˜π‘‹)) ∈ β„•0) β†’ (((πΆβ€˜(πΉβ€˜π‘‹)) β‰  0 β†’ (πΆβ€˜(πΉβ€˜π‘‹)) < (πΆβ€˜π‘‹)) ↔ (((πΆβ€˜π‘‹) β‰  0 β†’ (πΆβ€˜(πΉβ€˜π‘‹)) < (πΆβ€˜π‘‹)) ∧ ((πΆβ€˜π‘‹) = 0 β†’ (πΆβ€˜(πΉβ€˜π‘‹)) = 0))))
82, 6, 7syl2anc 411 1 (𝑋 ∈ 𝑆 β†’ (((πΆβ€˜(πΉβ€˜π‘‹)) β‰  0 β†’ (πΆβ€˜(πΉβ€˜π‘‹)) < (πΆβ€˜π‘‹)) ↔ (((πΆβ€˜π‘‹) β‰  0 β†’ (πΆβ€˜(πΉβ€˜π‘‹)) < (πΆβ€˜π‘‹)) ∧ ((πΆβ€˜π‘‹) = 0 β†’ (πΆβ€˜(πΉβ€˜π‘‹)) = 0))))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2146   β‰  wne 2345   class class class wbr 3998  βŸΆwf 5204  β€˜cfv 5208  0cc0 7786   < clt 7966  β„•0cn0 9147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-inn 8891  df-n0 9148  df-z 9225
This theorem is referenced by:  algcvga  12017
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