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Theorem cantnfub 43334
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 726 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑥 ∈ ran 𝐴) → 𝑀:𝑁⟶ω)
3 cantnfub.a . . . . . . . . 9 (𝜑𝐴:𝑁1-1𝑋)
43ad2antrr 726 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑥 ∈ ran 𝐴) → 𝐴:𝑁1-1𝑋)
5 f1f1orn 6859 . . . . . . . 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 7305 . . . . . . 7 ((𝐴:𝑁1-1-onto→ran 𝐴𝑥 ∈ ran 𝐴) → (𝐴𝑥) ∈ 𝑁)
86, 7sylancom 588 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑥 ∈ ran 𝐴) → (𝐴𝑥) ∈ 𝑁)
92, 8ffvelcdmd 7105 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑥 ∈ ran 𝐴) → (𝑀‘(𝐴𝑥)) ∈ ω)
10 peano1 7910 . . . . . 6 ∅ ∈ ω
1110a1i 11 . . . . 5 (((𝜑𝑥𝑋) ∧ ¬ 𝑥 ∈ ran 𝐴) → ∅ ∈ ω)
129, 11ifclda 4561 . . . 4 ((𝜑𝑥𝑋) → if(𝑥 ∈ ran 𝐴, (𝑀‘(𝐴𝑥)), ∅) ∈ ω)
13 cantnfub.f . . . 4 𝐹 = (𝑥𝑋 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(𝐴𝑥)), ∅))
1412, 13fmptd 7134 . . 3 (𝜑𝐹:𝑋⟶ω)
15 f1fn 6805 . . . . . . . 8 (𝐴:𝑁1-1𝑋𝐴 Fn 𝑁)
163, 15syl 17 . . . . . . 7 (𝜑𝐴 Fn 𝑁)
17 cantnfub.n . . . . . . . 8 (𝜑𝑁 ∈ ω)
18 nnon 7893 . . . . . . . . 9 (𝑁 ∈ ω → 𝑁 ∈ On)
19 onfin 9267 . . . . . . . . 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 511 . . . . . 6 (𝜑 → (𝐴 Fn 𝑁𝑁 ∈ Fin))
23 fnfi 9218 . . . . . 6 ((𝐴 Fn 𝑁𝑁 ∈ Fin) → 𝐴 ∈ Fin)
24 rnfi 9380 . . . . . 6 (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
2522, 23, 243syl 18 . . . . 5 (𝜑 → ran 𝐴 ∈ Fin)
26 eldifi 4131 . . . . . . . . 9 (𝑦 ∈ (𝑋 ∖ ran 𝐴) → 𝑦𝑋)
2726adantl 481 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → 𝑦𝑋)
28 eleq1w 2824 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ ran 𝐴𝑦 ∈ ran 𝐴))
29 2fveq3 6911 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑀‘(𝐴𝑥)) = (𝑀‘(𝐴𝑦)))
3028, 29ifbieq1d 4550 . . . . . . . . 9 (𝑥 = 𝑦 → if(𝑥 ∈ ran 𝐴, (𝑀‘(𝐴𝑥)), ∅) = if(𝑦 ∈ ran 𝐴, (𝑀‘(𝐴𝑦)), ∅))
31 fvex 6919 . . . . . . . . . 10 (𝑀‘(𝐴𝑦)) ∈ V
32 0ex 5307 . . . . . . . . . 10 ∅ ∈ V
3331, 32ifex 4576 . . . . . . . . 9 if(𝑦 ∈ ran 𝐴, (𝑀‘(𝐴𝑦)), ∅) ∈ V
3430, 13, 33fvmpt 7016 . . . . . . . 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 4132 . . . . . . . . 9 (𝑦 ∈ (𝑋 ∖ ran 𝐴) → ¬ 𝑦 ∈ ran 𝐴)
3736adantl 481 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → ¬ 𝑦 ∈ ran 𝐴)
3837iffalsed 4536 . . . . . . 7 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → if(𝑦 ∈ ran 𝐴, (𝑀‘(𝐴𝑦)), ∅) = ∅)
3935, 38eqtrd 2777 . . . . . 6 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → (𝐹𝑦) = ∅)
4014, 39suppss 8219 . . . . 5 (𝜑 → (𝐹 supp ∅) ⊆ ran 𝐴)
4125, 40ssfid 9301 . . . 4 (𝜑 → (𝐹 supp ∅) ∈ Fin)
4214ffund 6740 . . . . 5 (𝜑 → Fun 𝐹)
43 omelon 9686 . . . . . . . 8 ω ∈ On
4443a1i 11 . . . . . . 7 (𝜑 → ω ∈ On)
45 cantnfub.0 . . . . . . 7 (𝜑𝑋 ∈ On)
4644, 45elmapd 8880 . . . . . 6 (𝜑 → (𝐹 ∈ (ω ↑m 𝑋) ↔ 𝐹:𝑋⟶ω))
4714, 46mpbird 257 . . . . 5 (𝜑𝐹 ∈ (ω ↑m 𝑋))
4810a1i 11 . . . . 5 (𝜑 → ∅ ∈ ω)
49 funisfsupp 9407 . . . . 5 ((Fun 𝐹𝐹 ∈ (ω ↑m 𝑋) ∧ ∅ ∈ ω) → (𝐹 finSupp ∅ ↔ (𝐹 supp ∅) ∈ Fin))
5042, 47, 48, 49syl3anc 1373 . . . 4 (𝜑 → (𝐹 finSupp ∅ ↔ (𝐹 supp ∅) ∈ Fin))
5141, 50mpbird 257 . . 3 (𝜑𝐹 finSupp ∅)
52 eqid 2737 . . . 4 dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
5352, 44, 45cantnfs 9706 . . 3 (𝜑 → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
5414, 51, 53mpbir2and 713 . 2 (𝜑𝐹 ∈ dom (ω CNF 𝑋))
5552, 44, 45cantnff 9714 . . 3 (𝜑 → (ω CNF 𝑋):dom (ω CNF 𝑋)⟶(ω ↑o 𝑋))
5655, 54ffvelcdmd 7105 . 2 (𝜑 → ((ω CNF 𝑋)‘𝐹) ∈ (ω ↑o 𝑋))
5754, 56jca 511 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 206  wa 395   = wceq 1540  wcel 2108  cdif 3948  c0 4333  ifcif 4525   class class class wbr 5143  cmpt 5225  ccnv 5684  dom cdm 5685  ran crn 5686  Oncon0 6384  Fun wfun 6555   Fn wfn 6556  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  ωcom 7887   supp csupp 8185  o coe 8505  m cmap 8866  Fincfn 8985   finSupp cfsupp 9401   CNF ccnf 9701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-cnf 9702
This theorem is referenced by:  cantnfub2  43335
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