Theorem cantnfub 44513

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 𝑋) when (𝐴‘𝑛) is less than 𝑋 and (𝑀‘𝑛) is less than ω. Lemma 5.2 of [Schloeder] p. 15. (Contributed by RP, 31-Jan-2025.)

Hypotheses

Ref	Expression
cantnfub.0	⊢ (𝜑 → 𝑋 ∈ On)
cantnfub.n	⊢ (𝜑 → 𝑁 ∈ ω)
cantnfub.a	⊢ (𝜑 → 𝐴:𝑁⟶1-1𝑋)
cantnfub.m	⊢ (𝜑 → 𝑀:𝑁⟶ω)
cantnfub.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑥)), ∅))

Assertion

Ref	Expression
cantnfub	⊢ (𝜑 → (𝐹 ∈ dom (ω CNF 𝑋) ∧ ((ω CNF 𝑋)‘𝐹) ∈ (ω ↑o 𝑋)))

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

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

Proof of Theorem cantnfub

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfub.m	. . . . . . 7 ⊢ (𝜑 → 𝑀:𝑁⟶ω)
2	1	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 ∈ ran 𝐴) → 𝑀:𝑁⟶ω)
3		cantnfub.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴:𝑁⟶1-1𝑋)
4	3	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 ∈ ran 𝐴) → 𝐴:𝑁⟶1-1𝑋)
5		f1f1orn 6855	. . . . . . . 8 ⊢ (𝐴:𝑁⟶1-1𝑋 → 𝐴:𝑁⟶1-1ontoran 𝐴)
6	4, 5	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 ∈ ran 𝐴) → 𝐴:𝑁⟶1-1ontoran 𝐴)
7		f1ocnvdm 7313	. . . . . . 7 ⊢ ((𝐴:𝑁⟶1-1ontoran 𝐴 ∧ 𝑥 ∈ ran 𝐴) → (◡𝐴‘𝑥) ∈ 𝑁)
8	6, 7	sylancom 600	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 ∈ ran 𝐴) → (◡𝐴‘𝑥) ∈ 𝑁)
9	2, 8	ffvelcdmd 7106	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 ∈ ran 𝐴) → (𝑀‘(◡𝐴‘𝑥)) ∈ ω)
10		peano1 7921	. . . . . 6 ⊢ ∅ ∈ ω
11	10	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ ran 𝐴) → ∅ ∈ ω)
12	9, 11	ifclda 4521	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → if(𝑥 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑥)), ∅) ∈ ω)
13		cantnfub.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑥)), ∅))
14	12, 13	fmptd 7135	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶ω)
15		f1fn 6798	. . . . . . . 8 ⊢ (𝐴:𝑁⟶1-1𝑋 → 𝐴 Fn 𝑁)
16	3, 15	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐴 Fn 𝑁)
17		cantnfub.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ω)
18		nnon 7904	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
19		onfin 9255	. . . . . . . . 9 ⊢ (𝑁 ∈ On → (𝑁 ∈ Fin ↔ 𝑁 ∈ ω))
20	17, 18, 19	3syl 19	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ Fin ↔ 𝑁 ∈ ω))
21	17, 20	mpbird 260	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Fin)
22	16, 21	jca 521	. . . . . 6 ⊢ (𝜑 → (𝐴 Fn 𝑁 ∧ 𝑁 ∈ Fin))
23		fnfi 9218	. . . . . 6 ⊢ ((𝐴 Fn 𝑁 ∧ 𝑁 ∈ Fin) → 𝐴 ∈ Fin)
24		rnfi 9354	. . . . . 6 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
25	22, 23, 24	3syl 19	. . . . 5 ⊢ (𝜑 → ran 𝐴 ∈ Fin)
26		eldifi 4070	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑋 ∖ ran 𝐴) → 𝑦 ∈ 𝑋)
27	26	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → 𝑦 ∈ 𝑋)
28		eleq1w 2847	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ran 𝐴 ↔ 𝑦 ∈ ran 𝐴))
29		2fveq3 6910	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑀‘(◡𝐴‘𝑥)) = (𝑀‘(◡𝐴‘𝑦)))
30	28, 29	ifbieq1d 4510	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑥)), ∅) = if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅))
31		fvex 6918	. . . . . . . . . 10 ⊢ (𝑀‘(◡𝐴‘𝑦)) ∈ V
32		0ex 5264	. . . . . . . . . 10 ⊢ ∅ ∈ V
33	31, 32	ifex 4536	. . . . . . . . 9 ⊢ if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅) ∈ V
34	30, 13, 33	fvmpt 7013	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅))
35	27, 34	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → (𝐹‘𝑦) = if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅))
36		eldifn 4071	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑋 ∖ ran 𝐴) → ¬ 𝑦 ∈ ran 𝐴)
37	36	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → ¬ 𝑦 ∈ ran 𝐴)
38	37	iffalsed 4496	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅) = ∅)
39	35, 38	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → (𝐹‘𝑦) = ∅)
40	14, 39	suppss 8236	. . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ ran 𝐴)
41	25, 40	ssfid 9285	. . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ∈ Fin)
42	14	ffund 6733	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
43		omelon 9671	. . . . . . . 8 ⊢ ω ∈ On
44	43	a1i 11	. . . . . . 7 ⊢ (𝜑 → ω ∈ On)
45		cantnfub.0	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ On)
46	44, 45	elmapd 8885	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (ω ↑m 𝑋) ↔ 𝐹:𝑋⟶ω))
47	14, 46	mpbird 260	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ω ↑m 𝑋))
48	10	a1i 11	. . . . 5 ⊢ (𝜑 → ∅ ∈ ω)
49		funisfsupp 9384	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ (ω ↑m 𝑋) ∧ ∅ ∈ ω) → (𝐹 finSupp ∅ ↔ (𝐹 supp ∅) ∈ Fin))
50	42, 47, 48, 49	syl3anc 1399	. . . 4 ⊢ (𝜑 → (𝐹 finSupp ∅ ↔ (𝐹 supp ∅) ∈ Fin))
51	41, 50	mpbird 260	. . 3 ⊢ (𝜑 → 𝐹 finSupp ∅)
52		eqid 2763	. . . 4 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
53	52, 44, 45	cantnfs 9691	. . 3 ⊢ (𝜑 → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
54	14, 51, 53	mpbir2and 726	. 2 ⊢ (𝜑 → 𝐹 ∈ dom (ω CNF 𝑋))
55	52, 44, 45	cantnff 9699	. . 3 ⊢ (𝜑 → (ω CNF 𝑋):dom (ω CNF 𝑋)⟶(ω ↑o 𝑋))
56	55, 54	ffvelcdmd 7106	. 2 ⊢ (𝜑 → ((ω CNF 𝑋)‘𝐹) ∈ (ω ↑o 𝑋))
57	54, 56	jca 521	1 ⊢ (𝜑 → (𝐹 ∈ dom (ω CNF 𝑋) ∧ ((ω CNF 𝑋)‘𝐹) ∈ (ω ↑o 𝑋)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ∖ cdif 4048  ∅c0 4281  ifcif 4485   class class class wbr 5106   ↦ cmpt 5189  ^◡ccnv 5654  dom cdm 5655  ran crn 5656  Oncon0 6377  Fun wfun 6549   Fn wfn 6550  ⟶wf 6551  ⟶1-1wf1 6552  ⟶1-1ontowf1o 6554  ‘cfv 6555  (class class class)co 7440  ωcom 7898   supp csupp 8201   ↑_o coe 8500   ↑_m cmap 8872  Fincfn 8998   finSupp cfsupp 9378   CNF ccnf 9686

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-inf2 9666

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-supp 8202  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-seqom 8483  df-1o 8501  df-2o 8502  df-oadd 8505  df-omul 8506  df-oexp 8507  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-fsupp 9379  df-oi 9528  df-cnf 9687

This theorem is used by:  cantnfub2  44514