Theorem cauappcvgprlemm 7301

Description: Lemma for cauappcvgpr 7318. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)

Hypotheses

Ref	Expression
cauappcvgpr.f	⊢ (𝜑 → 𝐹:Q⟶Q)
cauappcvgpr.app	⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd	⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
cauappcvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩

Assertion

Ref	Expression
cauappcvgprlemm	⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))

Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐿,𝑟,𝑠   𝐴,𝑠,𝑝   𝐹,𝑙,𝑢,𝑝,𝑞,𝑟,𝑠   𝜑,𝑟,𝑠

Allowed substitution hints:   𝜑(𝑢,𝑙)   𝐴(𝑢,𝑟,𝑞,𝑙)   𝐿(𝑢,𝑙)

Proof of Theorem cauappcvgprlemm

Step	Hyp	Ref	Expression
1		fveq2 5340	. . . . . . 7 ⊢ (𝑝 = 1Q → (𝐹‘𝑝) = (𝐹‘1Q))
2	1	breq2d 3879	. . . . . 6 ⊢ (𝑝 = 1Q → (𝐴 <Q (𝐹‘𝑝) ↔ 𝐴 <Q (𝐹‘1Q)))
3		cauappcvgpr.bnd	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
4		1nq 7022	. . . . . . 7 ⊢ 1Q ∈ Q
5	4	a1i 9	. . . . . 6 ⊢ (𝜑 → 1Q ∈ Q)
6	2, 3, 5	rspcdva 2741	. . . . 5 ⊢ (𝜑 → 𝐴 <Q (𝐹‘1Q))
7		ltrelnq 7021	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
8	7	brel 4519	. . . . . 6 ⊢ (𝐴 <Q (𝐹‘1Q) → (𝐴 ∈ Q ∧ (𝐹‘1Q) ∈ Q))
9	8	simpld 111	. . . . 5 ⊢ (𝐴 <Q (𝐹‘1Q) → 𝐴 ∈ Q)
10	6, 9	syl 14	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Q)
11		halfnqq 7066	. . . 4 ⊢ (𝐴 ∈ Q → ∃𝑠 ∈ Q (𝑠 +Q 𝑠) = 𝐴)
12	10, 11	syl 14	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ Q (𝑠 +Q 𝑠) = 𝐴)
13		simplr 498	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ Q)
14	3	ad2antrr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
15		fveq2 5340	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑠 → (𝐹‘𝑝) = (𝐹‘𝑠))
16	15	breq2d 3879	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑠 → (𝐴 <Q (𝐹‘𝑝) ↔ 𝐴 <Q (𝐹‘𝑠)))
17	16	rspcv 2732	. . . . . . . . . 10 ⊢ (𝑠 ∈ Q → (∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝) → 𝐴 <Q (𝐹‘𝑠)))
18	17	ad2antlr 474	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → (∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝) → 𝐴 <Q (𝐹‘𝑠)))
19	14, 18	mpd 13	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝐴 <Q (𝐹‘𝑠))
20		breq1 3870	. . . . . . . . 9 ⊢ ((𝑠 +Q 𝑠) = 𝐴 → ((𝑠 +Q 𝑠) <Q (𝐹‘𝑠) ↔ 𝐴 <Q (𝐹‘𝑠)))
21	20	adantl 272	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ((𝑠 +Q 𝑠) <Q (𝐹‘𝑠) ↔ 𝐴 <Q (𝐹‘𝑠)))
22	19, 21	mpbird 166	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → (𝑠 +Q 𝑠) <Q (𝐹‘𝑠))
23		oveq2 5698	. . . . . . . . 9 ⊢ (𝑞 = 𝑠 → (𝑠 +Q 𝑞) = (𝑠 +Q 𝑠))
24		fveq2 5340	. . . . . . . . 9 ⊢ (𝑞 = 𝑠 → (𝐹‘𝑞) = (𝐹‘𝑠))
25	23, 24	breq12d 3880	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → ((𝑠 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑠) <Q (𝐹‘𝑠)))
26	25	rspcev 2736	. . . . . . 7 ⊢ ((𝑠 ∈ Q ∧ (𝑠 +Q 𝑠) <Q (𝐹‘𝑠)) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
27	13, 22, 26	syl2anc 404	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
28		oveq1 5697	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
29	28	breq1d 3877	. . . . . . . 8 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
30	29	rexbidv 2392	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
31		cauappcvgpr.lim	. . . . . . . . 9 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩
32	31	fveq2i 5343	. . . . . . . 8 ⊢ (1st ‘𝐿) = (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩)
33		nqex 7019	. . . . . . . . . 10 ⊢ Q ∈ V
34	33	rabex 4004	. . . . . . . . 9 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
35	33	rabex 4004	. . . . . . . . 9 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
36	34, 35	op1st 5955	. . . . . . . 8 ⊢ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
37	32, 36	eqtri 2115	. . . . . . 7 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
38	30, 37	elrab2 2788	. . . . . 6 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
39	13, 27, 38	sylanbrc 409	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ (1st ‘𝐿))
40	39	ex 114	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ Q) → ((𝑠 +Q 𝑠) = 𝐴 → 𝑠 ∈ (1st ‘𝐿)))
41	40	reximdva 2487	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ Q (𝑠 +Q 𝑠) = 𝐴 → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿)))
42	12, 41	mpd 13	. 2 ⊢ (𝜑 → ∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿))
43		cauappcvgpr.f	. . . . . 6 ⊢ (𝜑 → 𝐹:Q⟶Q)
44	43, 5	ffvelrnd 5474	. . . . 5 ⊢ (𝜑 → (𝐹‘1Q) ∈ Q)
45		addclnq 7031	. . . . 5 ⊢ (((𝐹‘1Q) ∈ Q ∧ 1Q ∈ Q) → ((𝐹‘1Q) +Q 1Q) ∈ Q)
46	44, 5, 45	syl2anc 404	. . . 4 ⊢ (𝜑 → ((𝐹‘1Q) +Q 1Q) ∈ Q)
47		addclnq 7031	. . . 4 ⊢ ((((𝐹‘1Q) +Q 1Q) ∈ Q ∧ 1Q ∈ Q) → (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ Q)
48	46, 5, 47	syl2anc 404	. . 3 ⊢ (𝜑 → (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ Q)
49		ltaddnq 7063	. . . . . 6 ⊢ ((((𝐹‘1Q) +Q 1Q) ∈ Q ∧ 1Q ∈ Q) → ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
50	46, 5, 49	syl2anc 404	. . . . 5 ⊢ (𝜑 → ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
51		fveq2 5340	. . . . . . . 8 ⊢ (𝑞 = 1Q → (𝐹‘𝑞) = (𝐹‘1Q))
52		id 19	. . . . . . . 8 ⊢ (𝑞 = 1Q → 𝑞 = 1Q)
53	51, 52	oveq12d 5708	. . . . . . 7 ⊢ (𝑞 = 1Q → ((𝐹‘𝑞) +Q 𝑞) = ((𝐹‘1Q) +Q 1Q))
54	53	breq1d 3877	. . . . . 6 ⊢ (𝑞 = 1Q → (((𝐹‘𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q) ↔ ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)))
55	54	rspcev 2736	. . . . 5 ⊢ ((1Q ∈ Q ∧ ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)) → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
56	5, 50, 55	syl2anc 404	. . . 4 ⊢ (𝜑 → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
57		breq2 3871	. . . . . 6 ⊢ (𝑢 = (((𝐹‘1Q) +Q 1Q) +Q 1Q) → (((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹‘𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)))
58	57	rexbidv 2392	. . . . 5 ⊢ (𝑢 = (((𝐹‘1Q) +Q 1Q) +Q 1Q) → (∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)))
59	31	fveq2i 5343	. . . . . 6 ⊢ (2nd ‘𝐿) = (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩)
60	34, 35	op2nd 5956	. . . . . 6 ⊢ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
61	59, 60	eqtri 2115	. . . . 5 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
62	58, 61	elrab2 2788	. . . 4 ⊢ ((((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿) ↔ ((((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ Q ∧ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q)))
63	48, 56, 62	sylanbrc 409	. . 3 ⊢ (𝜑 → (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿))
64		eleq1 2157	. . . 4 ⊢ (𝑟 = (((𝐹‘1Q) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd ‘𝐿) ↔ (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿)))
65	64	rspcev 2736	. . 3 ⊢ (((((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ Q ∧ (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿)) → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))
66	48, 63, 65	syl2anc 404	. 2 ⊢ (𝜑 → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))
67	42, 66	jca 301	1 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296   ∈ wcel 1445  ∀wral 2370  ∃wrex 2371  {crab 2374  ⟨cop 3469   class class class wbr 3867  ⟶wf 5045  ‘cfv 5049  (class class class)co 5690  1^st c1st 5947  2^nd c2nd 5948  Qcnq 6936  1_Qc1q 6937   +_Q cplq 6938   <_Q cltq 6941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009

This theorem is referenced by:  cauappcvgprlemcl  7309