Theorem cauappcvgprlemm 7453

Description: Lemma for cauappcvgpr 7470. 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 5421	. . . . . . 7 ⊢ (𝑝 = 1Q → (𝐹‘𝑝) = (𝐹‘1Q))
2	1	breq2d 3941	. . . . . 6 ⊢ (𝑝 = 1Q → (𝐴 <Q (𝐹‘𝑝) ↔ 𝐴 <Q (𝐹‘1Q)))
3		cauappcvgpr.bnd	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
4		1nq 7174	. . . . . . 7 ⊢ 1Q ∈ Q
5	4	a1i 9	. . . . . 6 ⊢ (𝜑 → 1Q ∈ Q)
6	2, 3, 5	rspcdva 2794	. . . . 5 ⊢ (𝜑 → 𝐴 <Q (𝐹‘1Q))
7		ltrelnq 7173	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
8	7	brel 4591	. . . . . 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 7218	. . . 4 ⊢ (𝐴 ∈ Q → ∃𝑠 ∈ Q (𝑠 +Q 𝑠) = 𝐴)
12	10, 11	syl 14	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ Q (𝑠 +Q 𝑠) = 𝐴)
13		simplr 519	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ Q)
14	3	ad2antrr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
15		fveq2 5421	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑠 → (𝐹‘𝑝) = (𝐹‘𝑠))
16	15	breq2d 3941	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑠 → (𝐴 <Q (𝐹‘𝑝) ↔ 𝐴 <Q (𝐹‘𝑠)))
17	16	rspcv 2785	. . . . . . . . . 10 ⊢ (𝑠 ∈ Q → (∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝) → 𝐴 <Q (𝐹‘𝑠)))
18	17	ad2antlr 480	. . . . . . . . 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 3932	. . . . . . . . 9 ⊢ ((𝑠 +Q 𝑠) = 𝐴 → ((𝑠 +Q 𝑠) <Q (𝐹‘𝑠) ↔ 𝐴 <Q (𝐹‘𝑠)))
21	20	adantl 275	. . . . . . . 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 5782	. . . . . . . . 9 ⊢ (𝑞 = 𝑠 → (𝑠 +Q 𝑞) = (𝑠 +Q 𝑠))
24		fveq2 5421	. . . . . . . . 9 ⊢ (𝑞 = 𝑠 → (𝐹‘𝑞) = (𝐹‘𝑠))
25	23, 24	breq12d 3942	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → ((𝑠 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑠) <Q (𝐹‘𝑠)))
26	25	rspcev 2789	. . . . . . 7 ⊢ ((𝑠 ∈ Q ∧ (𝑠 +Q 𝑠) <Q (𝐹‘𝑠)) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
27	13, 22, 26	syl2anc 408	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
28		oveq1 5781	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
29	28	breq1d 3939	. . . . . . . 8 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
30	29	rexbidv 2438	. . . . . . 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 5424	. . . . . . . 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 7171	. . . . . . . . . 10 ⊢ Q ∈ V
34	33	rabex 4072	. . . . . . . . 9 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
35	33	rabex 4072	. . . . . . . . 9 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
36	34, 35	op1st 6044	. . . . . . . 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 2160	. . . . . . 7 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
38	30, 37	elrab2 2843	. . . . . 6 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
39	13, 27, 38	sylanbrc 413	. . . . 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 2534	. . 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 5556	. . . . 5 ⊢ (𝜑 → (𝐹‘1Q) ∈ Q)
45		addclnq 7183	. . . . 5 ⊢ (((𝐹‘1Q) ∈ Q ∧ 1Q ∈ Q) → ((𝐹‘1Q) +Q 1Q) ∈ Q)
46	44, 5, 45	syl2anc 408	. . . 4 ⊢ (𝜑 → ((𝐹‘1Q) +Q 1Q) ∈ Q)
47		addclnq 7183	. . . 4 ⊢ ((((𝐹‘1Q) +Q 1Q) ∈ Q ∧ 1Q ∈ Q) → (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ Q)
48	46, 5, 47	syl2anc 408	. . 3 ⊢ (𝜑 → (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ Q)
49		ltaddnq 7215	. . . . . 6 ⊢ ((((𝐹‘1Q) +Q 1Q) ∈ Q ∧ 1Q ∈ Q) → ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
50	46, 5, 49	syl2anc 408	. . . . 5 ⊢ (𝜑 → ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
51		fveq2 5421	. . . . . . . 8 ⊢ (𝑞 = 1Q → (𝐹‘𝑞) = (𝐹‘1Q))
52		id 19	. . . . . . . 8 ⊢ (𝑞 = 1Q → 𝑞 = 1Q)
53	51, 52	oveq12d 5792	. . . . . . 7 ⊢ (𝑞 = 1Q → ((𝐹‘𝑞) +Q 𝑞) = ((𝐹‘1Q) +Q 1Q))
54	53	breq1d 3939	. . . . . 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 2789	. . . . 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 408	. . . 4 ⊢ (𝜑 → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
57		breq2 3933	. . . . . 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 2438	. . . . 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 5424	. . . . . 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 6045	. . . . . 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 2160	. . . . 5 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
62	58, 61	elrab2 2843	. . . 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 413	. . 3 ⊢ (𝜑 → (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿))
64		eleq1 2202	. . . 4 ⊢ (𝑟 = (((𝐹‘1Q) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd ‘𝐿) ↔ (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿)))
65	64	rspcev 2789	. . 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 408	. 2 ⊢ (𝜑 → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))
67	42, 66	jca 304	1 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420  ⟨cop 3530   class class class wbr 3929  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  1^st c1st 6036  2^nd c2nd 6037  Qcnq 7088  1_Qc1q 7089   +_Q cplq 7090   <_Q cltq 7093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161

This theorem is referenced by:  cauappcvgprlemcl  7461