Theorem cantnfub2 43771

Description: Given a finite number of terms of the form ((ω ↑_o (𝐴‘𝑛)) ·_o (𝑀‘𝑛)) with distinct exponents, we may order them from largest to smallest and find the sum is less than (ω ↑_o suc ∪ ran 𝐴) when (𝑀‘𝑛) is less than ω. Lemma 5.2 of [Schloeder] p. 15. (Contributed by RP, 9-Feb-2025.)

Hypotheses

Ref	Expression
cantnfub2.n	⊢ (𝜑 → 𝑁 ∈ ω)
cantnfub2.a	⊢ (𝜑 → 𝐴:𝑁–1-1→On)
cantnfub2.m	⊢ (𝜑 → 𝑀:𝑁⟶ω)
cantnfub2.f	⊢ 𝐹 = (𝑥 ∈ suc ∪ ran 𝐴 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑥)), ∅))

Assertion

Ref	Expression
cantnfub2	⊢ (𝜑 → (suc ∪ ran 𝐴 ∈ On ∧ 𝐹 ∈ dom (ω CNF suc ∪ ran 𝐴) ∧ ((ω CNF suc ∪ ran 𝐴)‘𝐹) ∈ (ω ↑o suc ∪ ran 𝐴)))

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝑀

Allowed substitution hints:   𝐹(𝑥)   𝑁(𝑥)

Proof of Theorem cantnfub2

Step	Hyp	Ref	Expression
1		cantnfub2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑁–1-1→On)
2		f1fn 6732	. . . . . . 7 ⊢ (𝐴:𝑁–1-1→On → 𝐴 Fn 𝑁)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 Fn 𝑁)
4		cantnfub2.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ω)
5		nnfi 9096	. . . . . . 7 ⊢ (𝑁 ∈ ω → 𝑁 ∈ Fin)
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ Fin)
7		fnfi 9106	. . . . . 6 ⊢ ((𝐴 Fn 𝑁 ∧ 𝑁 ∈ Fin) → 𝐴 ∈ Fin)
8	3, 6, 7	syl2anc 585	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
9		rnfi 9244	. . . . 5 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐴 ∈ Fin)
11		f1f 6731	. . . . . 6 ⊢ (𝐴:𝑁–1-1→On → 𝐴:𝑁⟶On)
12	1, 11	syl 17	. . . . 5 ⊢ (𝜑 → 𝐴:𝑁⟶On)
13	12	frnd 6671	. . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ On)
14		ssonuni 7728	. . . 4 ⊢ (ran 𝐴 ∈ Fin → (ran 𝐴 ⊆ On → ∪ ran 𝐴 ∈ On))
15	10, 13, 14	sylc 65	. . 3 ⊢ (𝜑 → ∪ ran 𝐴 ∈ On)
16		onsuc 7758	. . 3 ⊢ (∪ ran 𝐴 ∈ On → suc ∪ ran 𝐴 ∈ On)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → suc ∪ ran 𝐴 ∈ On)
18		onsucuni 7773	. . . . 5 ⊢ (ran 𝐴 ⊆ On → ran 𝐴 ⊆ suc ∪ ran 𝐴)
19	13, 18	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ suc ∪ ran 𝐴)
20		f1ssr 6737	. . . 4 ⊢ ((𝐴:𝑁–1-1→On ∧ ran 𝐴 ⊆ suc ∪ ran 𝐴) → 𝐴:𝑁–1-1→suc ∪ ran 𝐴)
21	1, 19, 20	syl2anc 585	. . 3 ⊢ (𝜑 → 𝐴:𝑁–1-1→suc ∪ ran 𝐴)
22		cantnfub2.m	. . 3 ⊢ (𝜑 → 𝑀:𝑁⟶ω)
23		cantnfub2.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ suc ∪ ran 𝐴 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑥)), ∅))
24	17, 4, 21, 22, 23	cantnfub 43770	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (ω CNF suc ∪ ran 𝐴) ∧ ((ω CNF suc ∪ ran 𝐴)‘𝐹) ∈ (ω ↑o suc ∪ ran 𝐴)))
25		3anass 1095	. 2 ⊢ ((suc ∪ ran 𝐴 ∈ On ∧ 𝐹 ∈ dom (ω CNF suc ∪ ran 𝐴) ∧ ((ω CNF suc ∪ ran 𝐴)‘𝐹) ∈ (ω ↑o suc ∪ ran 𝐴)) ↔ (suc ∪ ran 𝐴 ∈ On ∧ (𝐹 ∈ dom (ω CNF suc ∪ ran 𝐴) ∧ ((ω CNF suc ∪ ran 𝐴)‘𝐹) ∈ (ω ↑o suc ∪ ran 𝐴))))
26	17, 24, 25	sylanbrc 584	1 ⊢ (𝜑 → (suc ∪ ran 𝐴 ∈ On ∧ 𝐹 ∈ dom (ω CNF suc ∪ ran 𝐴) ∧ ((ω CNF suc ∪ ran 𝐴)‘𝐹) ∈ (ω ↑o suc ∪ ran 𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  ∅c0 4274  ifcif 4467  ∪ cuni 4851   ↦ cmpt 5167  ^◡ccnv 5624  dom cdm 5625  ran crn 5626  Oncon0 6318  suc csuc 6320   Fn wfn 6488  ⟶wf 6489  –1-1→wf1 6490  ‘cfv 6493  (class class class)co 7361  ωcom 7811   ↑_o coe 8398  Fincfn 8887   CNF ccnf 9576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-inf2 9556

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-seqom 8381  df-1o 8399  df-2o 8400  df-oadd 8403  df-omul 8404  df-oexp 8405  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-oi 9419  df-cnf 9577

This theorem is referenced by: (None)