Theorem cantnfub2 43899

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 6761	. . . . . . 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 9136	. . . . . . 7 ⊢ (𝑁 ∈ ω → 𝑁 ∈ Fin)
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ Fin)
7		fnfi 9146	. . . . . 6 ⊢ ((𝐴 Fn 𝑁 ∧ 𝑁 ∈ Fin) → 𝐴 ∈ Fin)
8	3, 6, 7	syl2anc 593	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
9		rnfi 9283	. . . . 5 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐴 ∈ Fin)
11		f1f 6760	. . . . . 6 ⊢ (𝐴:𝑁–1-1→On → 𝐴:𝑁⟶On)
12	1, 11	syl 17	. . . . 5 ⊢ (𝜑 → 𝐴:𝑁⟶On)
13	12	frnd 6700	. . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ On)
14		ssonuni 7763	. . . 4 ⊢ (ran 𝐴 ∈ Fin → (ran 𝐴 ⊆ On → ∪ ran 𝐴 ∈ On))
15	10, 13, 14	sylc 65	. . 3 ⊢ (𝜑 → ∪ ran 𝐴 ∈ On)
16		onsuc 7793	. . 3 ⊢ (∪ ran 𝐴 ∈ On → suc ∪ ran 𝐴 ∈ On)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → suc ∪ ran 𝐴 ∈ On)
18		onsucuni 7808	. . . . 5 ⊢ (ran 𝐴 ⊆ On → ran 𝐴 ⊆ suc ∪ ran 𝐴)
19	13, 18	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ suc ∪ ran 𝐴)
20		f1ssr 6768	. . . 4 ⊢ ((𝐴:𝑁–1-1→On ∧ ran 𝐴 ⊆ suc ∪ ran 𝐴) → 𝐴:𝑁–1-1→suc ∪ ran 𝐴)
21	1, 19, 20	syl2anc 593	. . 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 43898	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (ω CNF suc ∪ ran 𝐴) ∧ ((ω CNF suc ∪ ran 𝐴)‘𝐹) ∈ (ω ↑o suc ∪ ran 𝐴)))
25		3anass 1106	. 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 592	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 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ⊆ wss 3904  ∅c0 4285  ifcif 4480  ∪ cuni 4865   ↦ cmpt 5181  ^◡ccnv 5646  dom cdm 5647  ran crn 5648  Oncon0 6346  suc csuc 6348   Fn wfn 6516  ⟶wf 6517  –1-1→wf1 6518  ‘cfv 6521  (class class class)co 7396  ωcom 7846   ↑_o coe 8436  Fincfn 8927   CNF ccnf 9616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-seqom 8419  df-1o 8437  df-2o 8438  df-oadd 8441  df-omul 8442  df-oexp 8443  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9308  df-oi 9458  df-cnf 9617

This theorem is referenced by: (None)