Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cantnfub Structured version   Visualization version   GIF version

Theorem cantnfub 42004
Description: Given a finite number of terms of the form ((ω ↑o (𝐴𝑛)) ·o (𝑀𝑛)) with distinct exponents, we may order them from largest to smallest and find the sum is less than (ω ↑o 𝑋) when (𝐴𝑛) is less than 𝑋 and (𝑀𝑛) is less than ω. Lemma 5.2 of [Schloeder] p. 15. (Contributed by RP, 31-Jan-2025.)
Hypotheses
Ref Expression
cantnfub.0 (𝜑𝑋 ∈ On)
cantnfub.n (𝜑𝑁 ∈ ω)
cantnfub.a (𝜑𝐴:𝑁1-1𝑋)
cantnfub.m (𝜑𝑀:𝑁⟶ω)
cantnfub.f 𝐹 = (𝑥𝑋 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(𝐴𝑥)), ∅))
Assertion
Ref Expression
cantnfub (𝜑 → (𝐹 ∈ dom (ω CNF 𝑋) ∧ ((ω CNF 𝑋)‘𝐹) ∈ (ω ↑o 𝑋)))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝑀   𝑥,𝑋
Allowed substitution hints:   𝐹(𝑥)   𝑁(𝑥)

Proof of Theorem cantnfub
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cantnfub.m . . . . . . 7 (𝜑𝑀:𝑁⟶ω)
21ad2antrr 725 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑥 ∈ ran 𝐴) → 𝑀:𝑁⟶ω)
3 cantnfub.a . . . . . . . . 9 (𝜑𝐴:𝑁1-1𝑋)
43ad2antrr 725 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑥 ∈ ran 𝐴) → 𝐴:𝑁1-1𝑋)
5 f1f1orn 6841 . . . . . . . 8 (𝐴:𝑁1-1𝑋𝐴:𝑁1-1-onto→ran 𝐴)
64, 5syl 17 . . . . . . 7 (((𝜑𝑥𝑋) ∧ 𝑥 ∈ ran 𝐴) → 𝐴:𝑁1-1-onto→ran 𝐴)
7 f1ocnvdm 7278 . . . . . . 7 ((𝐴:𝑁1-1-onto→ran 𝐴𝑥 ∈ ran 𝐴) → (𝐴𝑥) ∈ 𝑁)
86, 7sylancom 589 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑥 ∈ ran 𝐴) → (𝐴𝑥) ∈ 𝑁)
92, 8ffvelcdmd 7083 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑥 ∈ ran 𝐴) → (𝑀‘(𝐴𝑥)) ∈ ω)
10 peano1 7874 . . . . . 6 ∅ ∈ ω
1110a1i 11 . . . . 5 (((𝜑𝑥𝑋) ∧ ¬ 𝑥 ∈ ran 𝐴) → ∅ ∈ ω)
129, 11ifclda 4562 . . . 4 ((𝜑𝑥𝑋) → if(𝑥 ∈ ran 𝐴, (𝑀‘(𝐴𝑥)), ∅) ∈ ω)
13 cantnfub.f . . . 4 𝐹 = (𝑥𝑋 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(𝐴𝑥)), ∅))
1412, 13fmptd 7109 . . 3 (𝜑𝐹:𝑋⟶ω)
15 f1fn 6785 . . . . . . . 8 (𝐴:𝑁1-1𝑋𝐴 Fn 𝑁)
163, 15syl 17 . . . . . . 7 (𝜑𝐴 Fn 𝑁)
17 cantnfub.n . . . . . . . 8 (𝜑𝑁 ∈ ω)
18 nnon 7856 . . . . . . . . 9 (𝑁 ∈ ω → 𝑁 ∈ On)
19 onfin 9226 . . . . . . . . 9 (𝑁 ∈ On → (𝑁 ∈ Fin ↔ 𝑁 ∈ ω))
2017, 18, 193syl 18 . . . . . . . 8 (𝜑 → (𝑁 ∈ Fin ↔ 𝑁 ∈ ω))
2117, 20mpbird 257 . . . . . . 7 (𝜑𝑁 ∈ Fin)
2216, 21jca 513 . . . . . 6 (𝜑 → (𝐴 Fn 𝑁𝑁 ∈ Fin))
23 fnfi 9177 . . . . . 6 ((𝐴 Fn 𝑁𝑁 ∈ Fin) → 𝐴 ∈ Fin)
24 rnfi 9331 . . . . . 6 (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
2522, 23, 243syl 18 . . . . 5 (𝜑 → ran 𝐴 ∈ Fin)
26 eldifi 4125 . . . . . . . . 9 (𝑦 ∈ (𝑋 ∖ ran 𝐴) → 𝑦𝑋)
2726adantl 483 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → 𝑦𝑋)
28 eleq1w 2817 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ ran 𝐴𝑦 ∈ ran 𝐴))
29 2fveq3 6893 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑀‘(𝐴𝑥)) = (𝑀‘(𝐴𝑦)))
3028, 29ifbieq1d 4551 . . . . . . . . 9 (𝑥 = 𝑦 → if(𝑥 ∈ ran 𝐴, (𝑀‘(𝐴𝑥)), ∅) = if(𝑦 ∈ ran 𝐴, (𝑀‘(𝐴𝑦)), ∅))
31 fvex 6901 . . . . . . . . . 10 (𝑀‘(𝐴𝑦)) ∈ V
32 0ex 5306 . . . . . . . . . 10 ∅ ∈ V
3331, 32ifex 4577 . . . . . . . . 9 if(𝑦 ∈ ran 𝐴, (𝑀‘(𝐴𝑦)), ∅) ∈ V
3430, 13, 33fvmpt 6994 . . . . . . . 8 (𝑦𝑋 → (𝐹𝑦) = if(𝑦 ∈ ran 𝐴, (𝑀‘(𝐴𝑦)), ∅))
3527, 34syl 17 . . . . . . 7 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → (𝐹𝑦) = if(𝑦 ∈ ran 𝐴, (𝑀‘(𝐴𝑦)), ∅))
36 eldifn 4126 . . . . . . . . 9 (𝑦 ∈ (𝑋 ∖ ran 𝐴) → ¬ 𝑦 ∈ ran 𝐴)
3736adantl 483 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → ¬ 𝑦 ∈ ran 𝐴)
3837iffalsed 4538 . . . . . . 7 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → if(𝑦 ∈ ran 𝐴, (𝑀‘(𝐴𝑦)), ∅) = ∅)
3935, 38eqtrd 2773 . . . . . 6 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → (𝐹𝑦) = ∅)
4014, 39suppss 8174 . . . . 5 (𝜑 → (𝐹 supp ∅) ⊆ ran 𝐴)
4125, 40ssfid 9263 . . . 4 (𝜑 → (𝐹 supp ∅) ∈ Fin)
4214ffund 6718 . . . . 5 (𝜑 → Fun 𝐹)
43 omelon 9637 . . . . . . . 8 ω ∈ On
4443a1i 11 . . . . . . 7 (𝜑 → ω ∈ On)
45 cantnfub.0 . . . . . . 7 (𝜑𝑋 ∈ On)
4644, 45elmapd 8830 . . . . . 6 (𝜑 → (𝐹 ∈ (ω ↑m 𝑋) ↔ 𝐹:𝑋⟶ω))
4714, 46mpbird 257 . . . . 5 (𝜑𝐹 ∈ (ω ↑m 𝑋))
4810a1i 11 . . . . 5 (𝜑 → ∅ ∈ ω)
49 funisfsupp 9363 . . . . 5 ((Fun 𝐹𝐹 ∈ (ω ↑m 𝑋) ∧ ∅ ∈ ω) → (𝐹 finSupp ∅ ↔ (𝐹 supp ∅) ∈ Fin))
5042, 47, 48, 49syl3anc 1372 . . . 4 (𝜑 → (𝐹 finSupp ∅ ↔ (𝐹 supp ∅) ∈ Fin))
5141, 50mpbird 257 . . 3 (𝜑𝐹 finSupp ∅)
52 eqid 2733 . . . 4 dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
5352, 44, 45cantnfs 9657 . . 3 (𝜑 → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
5414, 51, 53mpbir2and 712 . 2 (𝜑𝐹 ∈ dom (ω CNF 𝑋))
5552, 44, 45cantnff 9665 . . 3 (𝜑 → (ω CNF 𝑋):dom (ω CNF 𝑋)⟶(ω ↑o 𝑋))
5655, 54ffvelcdmd 7083 . 2 (𝜑 → ((ω CNF 𝑋)‘𝐹) ∈ (ω ↑o 𝑋))
5754, 56jca 513 1 (𝜑 → (𝐹 ∈ dom (ω CNF 𝑋) ∧ ((ω CNF 𝑋)‘𝐹) ∈ (ω ↑o 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cdif 3944  c0 4321  ifcif 4527   class class class wbr 5147  cmpt 5230  ccnv 5674  dom cdm 5675  ran crn 5676  Oncon0 6361  Fun wfun 6534   Fn wfn 6535  wf 6536  1-1wf1 6537  1-1-ontowf1o 6539  cfv 6540  (class class class)co 7404  ωcom 7850   supp csupp 8141  o coe 8460  m cmap 8816  Fincfn 8935   finSupp cfsupp 9357   CNF ccnf 9652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-supp 8142  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-seqom 8443  df-1o 8461  df-2o 8462  df-oadd 8465  df-omul 8466  df-oexp 8467  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-oi 9501  df-cnf 9653
This theorem is referenced by:  cantnfub2  42005
  Copyright terms: Public domain W3C validator