Theorem cantnfub2 44109

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 6779	. . . . . . 7 ⊢ (𝐴:𝑁–1-1→On → 𝐴 Fn 𝑁)
3	1, 2	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐴 Fn 𝑁)
4		cantnfub2.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ω)
5		nnfi 9159	. . . . . . 7 ⊢ (𝑁 ∈ ω → 𝑁 ∈ Fin)
6	4, 5	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ Fin)
7		fnfi 9169	. . . . . 6 ⊢ ((𝐴 Fn 𝑁 ∧ 𝑁 ∈ Fin) → 𝐴 ∈ Fin)
8	3, 6, 7	syl2anc 596	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
9		rnfi 9304	. . . . 5 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
10	8, 9	syl 18	. . . 4 ⊢ (𝜑 → ran 𝐴 ∈ Fin)
11		f1f 6778	. . . . . 6 ⊢ (𝐴:𝑁–1-1→On → 𝐴:𝑁⟶On)
12	1, 11	syl 18	. . . . 5 ⊢ (𝜑 → 𝐴:𝑁⟶On)
13	12	frnd 6718	. . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ On)
14		ssonuni 7785	. . . 4 ⊢ (ran 𝐴 ∈ Fin → (ran 𝐴 ⊆ On → ∪ ran 𝐴 ∈ On))
15	10, 13, 14	sylc 66	. . 3 ⊢ (𝜑 → ∪ ran 𝐴 ∈ On)
16		onsuc 7815	. . 3 ⊢ (∪ ran 𝐴 ∈ On → suc ∪ ran 𝐴 ∈ On)
17	15, 16	syl 18	. 2 ⊢ (𝜑 → suc ∪ ran 𝐴 ∈ On)
18		onsucuni 7830	. . . . 5 ⊢ (ran 𝐴 ⊆ On → ran 𝐴 ⊆ suc ∪ ran 𝐴)
19	13, 18	syl 18	. . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ suc ∪ ran 𝐴)
20		f1ssr 6786	. . . 4 ⊢ ((𝐴:𝑁–1-1→On ∧ ran 𝐴 ⊆ suc ∪ ran 𝐴) → 𝐴:𝑁–1-1→suc ∪ ran 𝐴)
21	1, 19, 20	syl2anc 596	. . 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 44108	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (ω CNF suc ∪ ran 𝐴) ∧ ((ω CNF suc ∪ ran 𝐴)‘𝐹) ∈ (ω ↑o suc ∪ ran 𝐴)))
25		3anass 1111	. 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 595	1 ⊢ (𝜑 → (suc ∪ ran 𝐴 ∈ On ∧ 𝐹 ∈ dom (ω CNF suc ∪ ran 𝐴) ∧ ((ω CNF suc ∪ ran 𝐴)‘𝐹) ∈ (ω ↑o suc ∪ ran 𝐴)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906  ∅c0 4286  ifcif 4489  ∪ cuni 4874   ↦ cmpt 5194  ^◡ccnv 5662  dom cdm 5663  ran crn 5664  Oncon0 6364  suc csuc 6366   Fn wfn 6535  ⟶wf 6536  –1-1→wf1 6537  ‘cfv 6540  (class class class)co 7419  ωcom 7868   ↑_o coe 8458  Fincfn 8949   CNF ccnf 9637

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  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 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-seqom 8441  df-1o 8459  df-2o 8460  df-oadd 8463  df-omul 8464  df-oexp 8465  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fsupp 9329  df-oi 9479  df-cnf 9638

This theorem is used by: (None)