Theorem cantnfub 43317

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–1-1→𝑋)
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 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 ∈ ran 𝐴) → 𝑀:𝑁⟶ω)
3		cantnfub.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴:𝑁–1-1→𝑋)
4	3	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 ∈ ran 𝐴) → 𝐴:𝑁–1-1→𝑋)
5		f1f1orn 6814	. . . . . . . 8 ⊢ (𝐴:𝑁–1-1→𝑋 → 𝐴:𝑁–1-1-onto→ran 𝐴)
6	4, 5	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 ∈ ran 𝐴) → 𝐴:𝑁–1-1-onto→ran 𝐴)
7		f1ocnvdm 7263	. . . . . . 7 ⊢ ((𝐴:𝑁–1-1-onto→ran 𝐴 ∧ 𝑥 ∈ ran 𝐴) → (◡𝐴‘𝑥) ∈ 𝑁)
8	6, 7	sylancom 588	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 ∈ ran 𝐴) → (◡𝐴‘𝑥) ∈ 𝑁)
9	2, 8	ffvelcdmd 7060	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 ∈ ran 𝐴) → (𝑀‘(◡𝐴‘𝑥)) ∈ ω)
10		peano1 7868	. . . . . 6 ⊢ ∅ ∈ ω
11	10	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ ran 𝐴) → ∅ ∈ ω)
12	9, 11	ifclda 4527	. . . 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 7089	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶ω)
15		f1fn 6760	. . . . . . . 8 ⊢ (𝐴:𝑁–1-1→𝑋 → 𝐴 Fn 𝑁)
16	3, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 Fn 𝑁)
17		cantnfub.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ω)
18		nnon 7851	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
19		onfin 9185	. . . . . . . . 9 ⊢ (𝑁 ∈ On → (𝑁 ∈ Fin ↔ 𝑁 ∈ ω))
20	17, 18, 19	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ Fin ↔ 𝑁 ∈ ω))
21	17, 20	mpbird 257	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ Fin)
22	16, 21	jca 511	. . . . . 6 ⊢ (𝜑 → (𝐴 Fn 𝑁 ∧ 𝑁 ∈ Fin))
23		fnfi 9148	. . . . . 6 ⊢ ((𝐴 Fn 𝑁 ∧ 𝑁 ∈ Fin) → 𝐴 ∈ Fin)
24		rnfi 9298	. . . . . 6 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
25	22, 23, 24	3syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐴 ∈ Fin)
26		eldifi 4097	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑋 ∖ ran 𝐴) → 𝑦 ∈ 𝑋)
27	26	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → 𝑦 ∈ 𝑋)
28		eleq1w 2812	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ran 𝐴 ↔ 𝑦 ∈ ran 𝐴))
29		2fveq3 6866	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑀‘(◡𝐴‘𝑥)) = (𝑀‘(◡𝐴‘𝑦)))
30	28, 29	ifbieq1d 4516	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑥)), ∅) = if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅))
31		fvex 6874	. . . . . . . . . 10 ⊢ (𝑀‘(◡𝐴‘𝑦)) ∈ V
32		0ex 5265	. . . . . . . . . 10 ⊢ ∅ ∈ V
33	31, 32	ifex 4542	. . . . . . . . 9 ⊢ if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅) ∈ V
34	30, 13, 33	fvmpt 6971	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅))
35	27, 34	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → (𝐹‘𝑦) = if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅))
36		eldifn 4098	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑋 ∖ ran 𝐴) → ¬ 𝑦 ∈ ran 𝐴)
37	36	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → ¬ 𝑦 ∈ ran 𝐴)
38	37	iffalsed 4502	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅) = ∅)
39	35, 38	eqtrd 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → (𝐹‘𝑦) = ∅)
40	14, 39	suppss 8176	. . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ ran 𝐴)
41	25, 40	ssfid 9219	. . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ∈ Fin)
42	14	ffund 6695	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
43		omelon 9606	. . . . . . . 8 ⊢ ω ∈ On
44	43	a1i 11	. . . . . . 7 ⊢ (𝜑 → ω ∈ On)
45		cantnfub.0	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ On)
46	44, 45	elmapd 8816	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (ω ↑m 𝑋) ↔ 𝐹:𝑋⟶ω))
47	14, 46	mpbird 257	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ω ↑m 𝑋))
48	10	a1i 11	. . . . 5 ⊢ (𝜑 → ∅ ∈ ω)
49		funisfsupp 9325	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ (ω ↑m 𝑋) ∧ ∅ ∈ ω) → (𝐹 finSupp ∅ ↔ (𝐹 supp ∅) ∈ Fin))
50	42, 47, 48, 49	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐹 finSupp ∅ ↔ (𝐹 supp ∅) ∈ Fin))
51	41, 50	mpbird 257	. . 3 ⊢ (𝜑 → 𝐹 finSupp ∅)
52		eqid 2730	. . . 4 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
53	52, 44, 45	cantnfs 9626	. . 3 ⊢ (𝜑 → (𝐹 ∈ dom (ω CNF 𝑋) ↔ (𝐹:𝑋⟶ω ∧ 𝐹 finSupp ∅)))
54	14, 51, 53	mpbir2and 713	. 2 ⊢ (𝜑 → 𝐹 ∈ dom (ω CNF 𝑋))
55	52, 44, 45	cantnff 9634	. . 3 ⊢ (𝜑 → (ω CNF 𝑋):dom (ω CNF 𝑋)⟶(ω ↑o 𝑋))
56	55, 54	ffvelcdmd 7060	. 2 ⊢ (𝜑 → ((ω CNF 𝑋)‘𝐹) ∈ (ω ↑o 𝑋))
57	54, 56	jca 511	1 ⊢ (𝜑 → (𝐹 ∈ dom (ω CNF 𝑋) ∧ ((ω CNF 𝑋)‘𝐹) ∈ (ω ↑o 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3914  ∅c0 4299  ifcif 4491   class class class wbr 5110   ↦ cmpt 5191  ^◡ccnv 5640  dom cdm 5641  ran crn 5642  Oncon0 6335  Fun wfun 6508   Fn wfn 6509  ⟶wf 6510  –1-1→wf1 6511  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  ωcom 7845   supp csupp 8142   ↑_o coe 8436   ↑_m cmap 8802  Fincfn 8921   finSupp cfsupp 9319   CNF ccnf 9621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-seqom 8419  df-1o 8437  df-2o 8438  df-oadd 8441  df-omul 8442  df-oexp 8443  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-oi 9470  df-cnf 9622

This theorem is referenced by:  cantnfub2  43318