Theorem cantnfub2 43425

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 6720	. . . . . . 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 9077	. . . . . . 7 ⊢ (𝑁 ∈ ω → 𝑁 ∈ Fin)
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ Fin)
7		fnfi 9087	. . . . . 6 ⊢ ((𝐴 Fn 𝑁 ∧ 𝑁 ∈ Fin) → 𝐴 ∈ Fin)
8	3, 6, 7	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
9		rnfi 9224	. . . . 5 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐴 ∈ Fin)
11		f1f 6719	. . . . . 6 ⊢ (𝐴:𝑁–1-1→On → 𝐴:𝑁⟶On)
12	1, 11	syl 17	. . . . 5 ⊢ (𝜑 → 𝐴:𝑁⟶On)
13	12	frnd 6659	. . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ On)
14		ssonuni 7713	. . . 4 ⊢ (ran 𝐴 ∈ Fin → (ran 𝐴 ⊆ On → ∪ ran 𝐴 ∈ On))
15	10, 13, 14	sylc 65	. . 3 ⊢ (𝜑 → ∪ ran 𝐴 ∈ On)
16		onsuc 7743	. . 3 ⊢ (∪ ran 𝐴 ∈ On → suc ∪ ran 𝐴 ∈ On)
17	15, 16	syl 17	. 2 ⊢ (𝜑 → suc ∪ ran 𝐴 ∈ On)
18		onsucuni 7758	. . . . 5 ⊢ (ran 𝐴 ⊆ On → ran 𝐴 ⊆ suc ∪ ran 𝐴)
19	13, 18	syl 17	. . . 4 ⊢ (𝜑 → ran 𝐴 ⊆ suc ∪ ran 𝐴)
20		f1ssr 6725	. . . 4 ⊢ ((𝐴:𝑁–1-1→On ∧ ran 𝐴 ⊆ suc ∪ ran 𝐴) → 𝐴:𝑁–1-1→suc ∪ ran 𝐴)
21	1, 19, 20	syl2anc 584	. . 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 43424	. 2 ⊢ (𝜑 → (𝐹 ∈ dom (ω CNF suc ∪ ran 𝐴) ∧ ((ω CNF suc ∪ ran 𝐴)‘𝐹) ∈ (ω ↑o suc ∪ ran 𝐴)))
25		3anass 1094	. 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 583	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 1086   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ∅c0 4280  ifcif 4472  ∪ cuni 4856   ↦ cmpt 5170  ^◡ccnv 5613  dom cdm 5614  ran crn 5615  Oncon0 6306  suc csuc 6308   Fn wfn 6476  ⟶wf 6477  –1-1→wf1 6478  ‘cfv 6481  (class class class)co 7346  ωcom 7796   ↑_o coe 8384  Fincfn 8869   CNF ccnf 9551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-seqom 8367  df-1o 8385  df-2o 8386  df-oadd 8389  df-omul 8390  df-oexp 8391  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-oi 9396  df-cnf 9552

This theorem is referenced by: (None)