Theorem cantnfub 43311

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 6860	. . . . . . . 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 7305	. . . . . . 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 7105	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑥 ∈ ran 𝐴) → (𝑀‘(◡𝐴‘𝑥)) ∈ ω)
10		peano1 7911	. . . . . 6 ⊢ ∅ ∈ ω
11	10	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ ran 𝐴) → ∅ ∈ ω)
12	9, 11	ifclda 4566	. . . 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 7134	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶ω)
15		f1fn 6806	. . . . . . . 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 7893	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
19		onfin 9265	. . . . . . . . 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 9216	. . . . . 6 ⊢ ((𝐴 Fn 𝑁 ∧ 𝑁 ∈ Fin) → 𝐴 ∈ Fin)
24		rnfi 9378	. . . . . 6 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin)
25	22, 23, 24	3syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐴 ∈ Fin)
26		eldifi 4141	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑋 ∖ ran 𝐴) → 𝑦 ∈ 𝑋)
27	26	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → 𝑦 ∈ 𝑋)
28		eleq1w 2822	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ran 𝐴 ↔ 𝑦 ∈ ran 𝐴))
29		2fveq3 6912	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑀‘(◡𝐴‘𝑥)) = (𝑀‘(◡𝐴‘𝑦)))
30	28, 29	ifbieq1d 4555	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑥)), ∅) = if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅))
31		fvex 6920	. . . . . . . . . 10 ⊢ (𝑀‘(◡𝐴‘𝑦)) ∈ V
32		0ex 5313	. . . . . . . . . 10 ⊢ ∅ ∈ V
33	31, 32	ifex 4581	. . . . . . . . 9 ⊢ if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅) ∈ V
34	30, 13, 33	fvmpt 7016	. . . . . . . 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 4142	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑋 ∖ ran 𝐴) → ¬ 𝑦 ∈ ran 𝐴)
37	36	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → ¬ 𝑦 ∈ ran 𝐴)
38	37	iffalsed 4542	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → if(𝑦 ∈ ran 𝐴, (𝑀‘(◡𝐴‘𝑦)), ∅) = ∅)
39	35, 38	eqtrd 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑋 ∖ ran 𝐴)) → (𝐹‘𝑦) = ∅)
40	14, 39	suppss 8218	. . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ ran 𝐴)
41	25, 40	ssfid 9299	. . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ∈ Fin)
42	14	ffund 6741	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
43		omelon 9684	. . . . . . . 8 ⊢ ω ∈ On
44	43	a1i 11	. . . . . . 7 ⊢ (𝜑 → ω ∈ On)
45		cantnfub.0	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ On)
46	44, 45	elmapd 8879	. . . . . 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 9405	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ (ω ↑m 𝑋) ∧ ∅ ∈ ω) → (𝐹 finSupp ∅ ↔ (𝐹 supp ∅) ∈ Fin))
50	42, 47, 48, 49	syl3anc 1370	. . . 4 ⊢ (𝜑 → (𝐹 finSupp ∅ ↔ (𝐹 supp ∅) ∈ Fin))
51	41, 50	mpbird 257	. . 3 ⊢ (𝜑 → 𝐹 finSupp ∅)
52		eqid 2735	. . . 4 ⊢ dom (ω CNF 𝑋) = dom (ω CNF 𝑋)
53	52, 44, 45	cantnfs 9704	. . 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 9712	. . 3 ⊢ (𝜑 → (ω CNF 𝑋):dom (ω CNF 𝑋)⟶(ω ↑o 𝑋))
56	55, 54	ffvelcdmd 7105	. 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 1537   ∈ wcel 2106   ∖ cdif 3960  ∅c0 4339  ifcif 4531   class class class wbr 5148   ↦ cmpt 5231  ^◡ccnv 5688  dom cdm 5689  ran crn 5690  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  ⟶wf 6559  –1-1→wf1 6560  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431  ωcom 7887   supp csupp 8184   ↑_o coe 8504   ↑_m cmap 8865  Fincfn 8984   finSupp cfsupp 9399   CNF ccnf 9699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-cnf 9700

This theorem is referenced by:  cantnfub2  43312