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Theorem cantnfub 42526
Description: Given a finite number of terms of the form ((Ο‰ ↑o (π΄β€˜π‘›)) Β·o (π‘€β€˜π‘›)) with distinct exponents, we may order them from largest to smallest and find the sum is less than (Ο‰ ↑o 𝑋) when (π΄β€˜π‘›) is less than 𝑋 and (π‘€β€˜π‘›) is less than Ο‰. Lemma 5.2 of [Schloeder] p. 15. (Contributed by RP, 31-Jan-2025.)
Hypotheses
Ref Expression
cantnfub.0 (πœ‘ β†’ 𝑋 ∈ On)
cantnfub.n (πœ‘ β†’ 𝑁 ∈ Ο‰)
cantnfub.a (πœ‘ β†’ 𝐴:𝑁–1-1→𝑋)
cantnfub.m (πœ‘ β†’ 𝑀:π‘βŸΆΟ‰)
cantnfub.f 𝐹 = (π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ ran 𝐴, (π‘€β€˜(β—‘π΄β€˜π‘₯)), βˆ…))
Assertion
Ref Expression
cantnfub (πœ‘ β†’ (𝐹 ∈ dom (Ο‰ CNF 𝑋) ∧ ((Ο‰ CNF 𝑋)β€˜πΉ) ∈ (Ο‰ ↑o 𝑋)))
Distinct variable groups:   πœ‘,π‘₯   π‘₯,𝐴   π‘₯,𝑀   π‘₯,𝑋
Allowed substitution hints:   𝐹(π‘₯)   𝑁(π‘₯)

Proof of Theorem cantnfub
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cantnfub.m . . . . . . 7 (πœ‘ β†’ 𝑀:π‘βŸΆΟ‰)
21ad2antrr 723 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ π‘₯ ∈ ran 𝐴) β†’ 𝑀:π‘βŸΆΟ‰)
3 cantnfub.a . . . . . . . . 9 (πœ‘ β†’ 𝐴:𝑁–1-1→𝑋)
43ad2antrr 723 . . . . . . . 8 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ π‘₯ ∈ ran 𝐴) β†’ 𝐴:𝑁–1-1→𝑋)
5 f1f1orn 6834 . . . . . . . 8 (𝐴:𝑁–1-1→𝑋 β†’ 𝐴:𝑁–1-1-ontoβ†’ran 𝐴)
64, 5syl 17 . . . . . . 7 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ π‘₯ ∈ ran 𝐴) β†’ 𝐴:𝑁–1-1-ontoβ†’ran 𝐴)
7 f1ocnvdm 7275 . . . . . . 7 ((𝐴:𝑁–1-1-ontoβ†’ran 𝐴 ∧ π‘₯ ∈ ran 𝐴) β†’ (β—‘π΄β€˜π‘₯) ∈ 𝑁)
86, 7sylancom 587 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ π‘₯ ∈ ran 𝐴) β†’ (β—‘π΄β€˜π‘₯) ∈ 𝑁)
92, 8ffvelcdmd 7077 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ π‘₯ ∈ ran 𝐴) β†’ (π‘€β€˜(β—‘π΄β€˜π‘₯)) ∈ Ο‰)
10 peano1 7872 . . . . . 6 βˆ… ∈ Ο‰
1110a1i 11 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑋) ∧ Β¬ π‘₯ ∈ ran 𝐴) β†’ βˆ… ∈ Ο‰)
129, 11ifclda 4555 . . . 4 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ if(π‘₯ ∈ ran 𝐴, (π‘€β€˜(β—‘π΄β€˜π‘₯)), βˆ…) ∈ Ο‰)
13 cantnfub.f . . . 4 𝐹 = (π‘₯ ∈ 𝑋 ↦ if(π‘₯ ∈ ran 𝐴, (π‘€β€˜(β—‘π΄β€˜π‘₯)), βˆ…))
1412, 13fmptd 7105 . . 3 (πœ‘ β†’ 𝐹:π‘‹βŸΆΟ‰)
15 f1fn 6778 . . . . . . . 8 (𝐴:𝑁–1-1→𝑋 β†’ 𝐴 Fn 𝑁)
163, 15syl 17 . . . . . . 7 (πœ‘ β†’ 𝐴 Fn 𝑁)
17 cantnfub.n . . . . . . . 8 (πœ‘ β†’ 𝑁 ∈ Ο‰)
18 nnon 7854 . . . . . . . . 9 (𝑁 ∈ Ο‰ β†’ 𝑁 ∈ On)
19 onfin 9225 . . . . . . . . 9 (𝑁 ∈ On β†’ (𝑁 ∈ Fin ↔ 𝑁 ∈ Ο‰))
2017, 18, 193syl 18 . . . . . . . 8 (πœ‘ β†’ (𝑁 ∈ Fin ↔ 𝑁 ∈ Ο‰))
2117, 20mpbird 257 . . . . . . 7 (πœ‘ β†’ 𝑁 ∈ Fin)
2216, 21jca 511 . . . . . 6 (πœ‘ β†’ (𝐴 Fn 𝑁 ∧ 𝑁 ∈ Fin))
23 fnfi 9176 . . . . . 6 ((𝐴 Fn 𝑁 ∧ 𝑁 ∈ Fin) β†’ 𝐴 ∈ Fin)
24 rnfi 9330 . . . . . 6 (𝐴 ∈ Fin β†’ ran 𝐴 ∈ Fin)
2522, 23, 243syl 18 . . . . 5 (πœ‘ β†’ ran 𝐴 ∈ Fin)
26 eldifi 4118 . . . . . . . . 9 (𝑦 ∈ (𝑋 βˆ– ran 𝐴) β†’ 𝑦 ∈ 𝑋)
2726adantl 481 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ (𝑋 βˆ– ran 𝐴)) β†’ 𝑦 ∈ 𝑋)
28 eleq1w 2808 . . . . . . . . . 10 (π‘₯ = 𝑦 β†’ (π‘₯ ∈ ran 𝐴 ↔ 𝑦 ∈ ran 𝐴))
29 2fveq3 6886 . . . . . . . . . 10 (π‘₯ = 𝑦 β†’ (π‘€β€˜(β—‘π΄β€˜π‘₯)) = (π‘€β€˜(β—‘π΄β€˜π‘¦)))
3028, 29ifbieq1d 4544 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ if(π‘₯ ∈ ran 𝐴, (π‘€β€˜(β—‘π΄β€˜π‘₯)), βˆ…) = if(𝑦 ∈ ran 𝐴, (π‘€β€˜(β—‘π΄β€˜π‘¦)), βˆ…))
31 fvex 6894 . . . . . . . . . 10 (π‘€β€˜(β—‘π΄β€˜π‘¦)) ∈ V
32 0ex 5297 . . . . . . . . . 10 βˆ… ∈ V
3331, 32ifex 4570 . . . . . . . . 9 if(𝑦 ∈ ran 𝐴, (π‘€β€˜(β—‘π΄β€˜π‘¦)), βˆ…) ∈ V
3430, 13, 33fvmpt 6988 . . . . . . . 8 (𝑦 ∈ 𝑋 β†’ (πΉβ€˜π‘¦) = if(𝑦 ∈ ran 𝐴, (π‘€β€˜(β—‘π΄β€˜π‘¦)), βˆ…))
3527, 34syl 17 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (𝑋 βˆ– ran 𝐴)) β†’ (πΉβ€˜π‘¦) = if(𝑦 ∈ ran 𝐴, (π‘€β€˜(β—‘π΄β€˜π‘¦)), βˆ…))
36 eldifn 4119 . . . . . . . . 9 (𝑦 ∈ (𝑋 βˆ– ran 𝐴) β†’ Β¬ 𝑦 ∈ ran 𝐴)
3736adantl 481 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ (𝑋 βˆ– ran 𝐴)) β†’ Β¬ 𝑦 ∈ ran 𝐴)
3837iffalsed 4531 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (𝑋 βˆ– ran 𝐴)) β†’ if(𝑦 ∈ ran 𝐴, (π‘€β€˜(β—‘π΄β€˜π‘¦)), βˆ…) = βˆ…)
3935, 38eqtrd 2764 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (𝑋 βˆ– ran 𝐴)) β†’ (πΉβ€˜π‘¦) = βˆ…)
4014, 39suppss 8173 . . . . 5 (πœ‘ β†’ (𝐹 supp βˆ…) βŠ† ran 𝐴)
4125, 40ssfid 9262 . . . 4 (πœ‘ β†’ (𝐹 supp βˆ…) ∈ Fin)
4214ffund 6711 . . . . 5 (πœ‘ β†’ Fun 𝐹)
43 omelon 9636 . . . . . . . 8 Ο‰ ∈ On
4443a1i 11 . . . . . . 7 (πœ‘ β†’ Ο‰ ∈ On)
45 cantnfub.0 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ On)
4644, 45elmapd 8829 . . . . . 6 (πœ‘ β†’ (𝐹 ∈ (Ο‰ ↑m 𝑋) ↔ 𝐹:π‘‹βŸΆΟ‰))
4714, 46mpbird 257 . . . . 5 (πœ‘ β†’ 𝐹 ∈ (Ο‰ ↑m 𝑋))
4810a1i 11 . . . . 5 (πœ‘ β†’ βˆ… ∈ Ο‰)
49 funisfsupp 9362 . . . . 5 ((Fun 𝐹 ∧ 𝐹 ∈ (Ο‰ ↑m 𝑋) ∧ βˆ… ∈ Ο‰) β†’ (𝐹 finSupp βˆ… ↔ (𝐹 supp βˆ…) ∈ Fin))
5042, 47, 48, 49syl3anc 1368 . . . 4 (πœ‘ β†’ (𝐹 finSupp βˆ… ↔ (𝐹 supp βˆ…) ∈ Fin))
5141, 50mpbird 257 . . 3 (πœ‘ β†’ 𝐹 finSupp βˆ…)
52 eqid 2724 . . . 4 dom (Ο‰ CNF 𝑋) = dom (Ο‰ CNF 𝑋)
5352, 44, 45cantnfs 9656 . . 3 (πœ‘ β†’ (𝐹 ∈ dom (Ο‰ CNF 𝑋) ↔ (𝐹:π‘‹βŸΆΟ‰ ∧ 𝐹 finSupp βˆ…)))
5414, 51, 53mpbir2and 710 . 2 (πœ‘ β†’ 𝐹 ∈ dom (Ο‰ CNF 𝑋))
5552, 44, 45cantnff 9664 . . 3 (πœ‘ β†’ (Ο‰ CNF 𝑋):dom (Ο‰ CNF 𝑋)⟢(Ο‰ ↑o 𝑋))
5655, 54ffvelcdmd 7077 . 2 (πœ‘ β†’ ((Ο‰ CNF 𝑋)β€˜πΉ) ∈ (Ο‰ ↑o 𝑋))
5754, 56jca 511 1 (πœ‘ β†’ (𝐹 ∈ dom (Ο‰ CNF 𝑋) ∧ ((Ο‰ CNF 𝑋)β€˜πΉ) ∈ (Ο‰ ↑o 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   βˆ– cdif 3937  βˆ…c0 4314  ifcif 4520   class class class wbr 5138   ↦ cmpt 5221  β—‘ccnv 5665  dom cdm 5666  ran crn 5667  Oncon0 6354  Fun wfun 6527   Fn wfn 6528  βŸΆwf 6529  β€“1-1β†’wf1 6530  β€“1-1-ontoβ†’wf1o 6532  β€˜cfv 6533  (class class class)co 7401  Ο‰com 7848   supp csupp 8140   ↑o coe 8460   ↑m cmap 8815  Fincfn 8934   finSupp cfsupp 9356   CNF ccnf 9651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9631
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-supp 8141  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 8698  df-map 8817  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-fsupp 9357  df-oi 9500  df-cnf 9652
This theorem is referenced by:  cantnfub2  42527
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