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Theorem cantnfub 43311
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 6860 . . . . . . . 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 7911 . . . . . 6 ∅ ∈ ω
1110a1i 11 . . . . 5 (((𝜑𝑥𝑋) ∧ ¬ 𝑥 ∈ ran 𝐴) → ∅ ∈ ω)
129, 11ifclda 4566 . . . 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 6806 . . . . . . . 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 9265 . . . . . . . . 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 9216 . . . . . 6 ((𝐴 Fn 𝑁𝑁 ∈ Fin) → 𝐴 ∈ Fin)
24 rnfi 9378 . . . . . 6 (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
2522, 23, 243syl 18 . . . . 5 (𝜑 → ran 𝐴 ∈ Fin)
26 eldifi 4141 . . . . . . . . 9 (𝑦 ∈ (𝑋 ∖ ran 𝐴) → 𝑦𝑋)
2726adantl 481 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → 𝑦𝑋)
28 eleq1w 2822 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 ∈ ran 𝐴𝑦 ∈ ran 𝐴))
29 2fveq3 6912 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑀‘(𝐴𝑥)) = (𝑀‘(𝐴𝑦)))
3028, 29ifbieq1d 4555 . . . . . . . . 9 (𝑥 = 𝑦 → if(𝑥 ∈ ran 𝐴, (𝑀‘(𝐴𝑥)), ∅) = if(𝑦 ∈ ran 𝐴, (𝑀‘(𝐴𝑦)), ∅))
31 fvex 6920 . . . . . . . . . 10 (𝑀‘(𝐴𝑦)) ∈ V
32 0ex 5313 . . . . . . . . . 10 ∅ ∈ V
3331, 32ifex 4581 . . . . . . . . 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 4142 . . . . . . . . 9 (𝑦 ∈ (𝑋 ∖ ran 𝐴) → ¬ 𝑦 ∈ ran 𝐴)
3736adantl 481 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → ¬ 𝑦 ∈ ran 𝐴)
3837iffalsed 4542 . . . . . . 7 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → if(𝑦 ∈ ran 𝐴, (𝑀‘(𝐴𝑦)), ∅) = ∅)
3935, 38eqtrd 2775 . . . . . 6 ((𝜑𝑦 ∈ (𝑋 ∖ ran 𝐴)) → (𝐹𝑦) = ∅)
4014, 39suppss 8218 . . . . 5 (𝜑 → (𝐹 supp ∅) ⊆ ran 𝐴)
4125, 40ssfid 9299 . . . 4 (𝜑 → (𝐹 supp ∅) ∈ Fin)
4214ffund 6741 . . . . 5 (𝜑 → Fun 𝐹)
43 omelon 9684 . . . . . . . 8 ω ∈ On
4443a1i 11 . . . . . . 7 (𝜑 → ω ∈ On)
45 cantnfub.0 . . . . . . 7 (𝜑𝑋 ∈ On)
4644, 45elmapd 8879 . . . . . 6 (𝜑 → (𝐹 ∈ (ω ↑m 𝑋) ↔ 𝐹:𝑋⟶ω))
4714, 46mpbird 257 . . . . 5 (𝜑𝐹 ∈ (ω ↑m 𝑋))
4810a1i 11 . . . . 5 (𝜑 → ∅ ∈ ω)
49 funisfsupp 9405 . . . . 5 ((Fun 𝐹𝐹 ∈ (ω ↑m 𝑋) ∧ ∅ ∈ ω) → (𝐹 finSupp ∅ ↔ (𝐹 supp ∅) ∈ Fin))
5042, 47, 48, 49syl3anc 1370 . . . 4 (𝜑 → (𝐹 finSupp ∅ ↔ (𝐹 supp ∅) ∈ Fin))
5141, 50mpbird 257 . . 3 (𝜑𝐹 finSupp ∅)
52 eqid 2735 . . . 4 dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
5352, 44, 45cantnfs 9704 . . 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 9712 . . 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 1537  wcel 2106  cdif 3960  c0 4339  ifcif 4531   class class class wbr 5148  cmpt 5231  ccnv 5688  dom cdm 5689  ran crn 5690  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  wf 6559  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  ωcom 7887   supp csupp 8184  o coe 8504  m cmap 8865  Fincfn 8984   finSupp cfsupp 9399   CNF ccnf 9699
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  ax-inf2 9679
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-op 4638  df-uni 4913  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-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-cnf 9700
This theorem is referenced by:  cantnfub2  43312
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