Theorem cantnfub2 44164

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 6773	. . . . . . 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 9163	. . . . . . 7 ⊢ (𝑁 ∈ ω → 𝑁 ∈ Fin)
6	4, 5	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ Fin)
7		fnfi 9173	. . . . . 6 ⊢ ((𝐴 Fn 𝑁 ∧ 𝑁 ∈ Fin) → 𝐴 ∈ Fin)
8	3, 6, 7	syl2anc 596	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
9		rnfi 9308	. . . . 5 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
10	8, 9	syl 18	. . . 4 ⊢ (𝜑 → ran 𝐴 ∈ Fin)
11		f1f 6772	. . . . . 6 ⊢ (𝐴:𝑁–1-1→On → 𝐴:𝑁⟶On)
12	1, 11	syl 18	. . . . 5 ⊢ (𝜑 → 𝐴:𝑁⟶On)
13	12	frnd 6712	. . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ On)
14		ssonuni 7780	. . . 4 ⊢ (ran 𝐴 ∈ Fin → (ran 𝐴 ⊆ On → ∪ ran 𝐴 ∈ On))
15	10, 13, 14	sylc 66	. . 3 ⊢ (𝜑 → ∪ ran 𝐴 ∈ On)
16		onsuc 7810	. . 3 ⊢ (∪ ran 𝐴 ∈ On → suc ∪ ran 𝐴 ∈ On)
17	15, 16	syl 18	. 2 ⊢ (𝜑 → suc ∪ ran 𝐴 ∈ On)
18		onsucuni 7825	. . . . 5 ⊢ (ran 𝐴 ⊆ On → ran 𝐴 ⊆ suc ∪ ran 𝐴)
19	13, 18	syl 18	. . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ suc ∪ ran 𝐴)
20		f1ssr 6780	. . . 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 44163	. 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 2145   ⊆ wss 3899  ∅c0 4279  ifcif 4482  ∪ cuni 4867   ↦ cmpt 5186  ^◡ccnv 5654  dom cdm 5655  ran crn 5656  Oncon0 6357  suc csuc 6359   Fn wfn 6528  ⟶wf 6529  –1-1→wf1 6530  ‘cfv 6533  (class class class)co 7414  ωcom 7863   ↑_o coe 8455  Fincfn 8953   CNF ccnf 9641

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-inf2 9621

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  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 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-seqom 8438  df-1o 8456  df-2o 8457  df-oadd 8460  df-omul 8461  df-oexp 8462  df-map 8829  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-fsupp 9333  df-oi 9483  df-cnf 9642

This theorem is used by: (None)