Theorem cauappcvgprlemm 7908

Description: Lemma for cauappcvgpr 7925. 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 5648	. . . . . . 7 ⊢ (𝑝 = 1Q → (𝐹‘𝑝) = (𝐹‘1Q))
2	1	breq2d 4105	. . . . . 6 ⊢ (𝑝 = 1Q → (𝐴 <Q (𝐹‘𝑝) ↔ 𝐴 <Q (𝐹‘1Q)))
3		cauappcvgpr.bnd	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
4		1nq 7629	. . . . . . 7 ⊢ 1Q ∈ Q
5	4	a1i 9	. . . . . 6 ⊢ (𝜑 → 1Q ∈ Q)
6	2, 3, 5	rspcdva 2916	. . . . 5 ⊢ (𝜑 → 𝐴 <Q (𝐹‘1Q))
7		ltrelnq 7628	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
8	7	brel 4784	. . . . . 6 ⊢ (𝐴 <Q (𝐹‘1Q) → (𝐴 ∈ Q ∧ (𝐹‘1Q) ∈ Q))
9	8	simpld 112	. . . . 5 ⊢ (𝐴 <Q (𝐹‘1Q) → 𝐴 ∈ Q)
10	6, 9	syl 14	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Q)
11		halfnqq 7673	. . . 4 ⊢ (𝐴 ∈ Q → ∃𝑠 ∈ Q (𝑠 +Q 𝑠) = 𝐴)
12	10, 11	syl 14	. . 3 ⊢ (𝜑 → ∃𝑠 ∈ Q (𝑠 +Q 𝑠) = 𝐴)
13		simplr 529	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ Q)
14	3	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
15		fveq2 5648	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑠 → (𝐹‘𝑝) = (𝐹‘𝑠))
16	15	breq2d 4105	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑠 → (𝐴 <Q (𝐹‘𝑝) ↔ 𝐴 <Q (𝐹‘𝑠)))
17	16	rspcv 2907	. . . . . . . . . 10 ⊢ (𝑠 ∈ Q → (∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝) → 𝐴 <Q (𝐹‘𝑠)))
18	17	ad2antlr 489	. . . . . . . . 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 4096	. . . . . . . . 9 ⊢ ((𝑠 +Q 𝑠) = 𝐴 → ((𝑠 +Q 𝑠) <Q (𝐹‘𝑠) ↔ 𝐴 <Q (𝐹‘𝑠)))
21	20	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ((𝑠 +Q 𝑠) <Q (𝐹‘𝑠) ↔ 𝐴 <Q (𝐹‘𝑠)))
22	19, 21	mpbird 167	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → (𝑠 +Q 𝑠) <Q (𝐹‘𝑠))
23		oveq2 6036	. . . . . . . . 9 ⊢ (𝑞 = 𝑠 → (𝑠 +Q 𝑞) = (𝑠 +Q 𝑠))
24		fveq2 5648	. . . . . . . . 9 ⊢ (𝑞 = 𝑠 → (𝐹‘𝑞) = (𝐹‘𝑠))
25	23, 24	breq12d 4106	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → ((𝑠 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑠) <Q (𝐹‘𝑠)))
26	25	rspcev 2911	. . . . . . 7 ⊢ ((𝑠 ∈ Q ∧ (𝑠 +Q 𝑠) <Q (𝐹‘𝑠)) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
27	13, 22, 26	syl2anc 411	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞))
28		oveq1 6035	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
29	28	breq1d 4103	. . . . . . . 8 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
30	29	rexbidv 2534	. . . . . . 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 5651	. . . . . . . 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 7626	. . . . . . . . . 10 ⊢ Q ∈ V
34	33	rabex 4239	. . . . . . . . 9 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
35	33	rabex 4239	. . . . . . . . 9 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
36	34, 35	op1st 6318	. . . . . . . 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 2252	. . . . . . 7 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
38	30, 37	elrab2 2966	. . . . . 6 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑞 ∈ Q (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
39	13, 27, 38	sylanbrc 417	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ Q) ∧ (𝑠 +Q 𝑠) = 𝐴) → 𝑠 ∈ (1st ‘𝐿))
40	39	ex 115	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ Q) → ((𝑠 +Q 𝑠) = 𝐴 → 𝑠 ∈ (1st ‘𝐿)))
41	40	reximdva 2635	. . 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	ffvelcdmd 5791	. . . . 5 ⊢ (𝜑 → (𝐹‘1Q) ∈ Q)
45		addclnq 7638	. . . . 5 ⊢ (((𝐹‘1Q) ∈ Q ∧ 1Q ∈ Q) → ((𝐹‘1Q) +Q 1Q) ∈ Q)
46	44, 5, 45	syl2anc 411	. . . 4 ⊢ (𝜑 → ((𝐹‘1Q) +Q 1Q) ∈ Q)
47		addclnq 7638	. . . 4 ⊢ ((((𝐹‘1Q) +Q 1Q) ∈ Q ∧ 1Q ∈ Q) → (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ Q)
48	46, 5, 47	syl2anc 411	. . 3 ⊢ (𝜑 → (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ Q)
49		ltaddnq 7670	. . . . . 6 ⊢ ((((𝐹‘1Q) +Q 1Q) ∈ Q ∧ 1Q ∈ Q) → ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
50	46, 5, 49	syl2anc 411	. . . . 5 ⊢ (𝜑 → ((𝐹‘1Q) +Q 1Q) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
51		fveq2 5648	. . . . . . . 8 ⊢ (𝑞 = 1Q → (𝐹‘𝑞) = (𝐹‘1Q))
52		id 19	. . . . . . . 8 ⊢ (𝑞 = 1Q → 𝑞 = 1Q)
53	51, 52	oveq12d 6046	. . . . . . 7 ⊢ (𝑞 = 1Q → ((𝐹‘𝑞) +Q 𝑞) = ((𝐹‘1Q) +Q 1Q))
54	53	breq1d 4103	. . . . . 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 2911	. . . . 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 411	. . . 4 ⊢ (𝜑 → ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q (((𝐹‘1Q) +Q 1Q) +Q 1Q))
57		breq2 4097	. . . . . 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 2534	. . . . 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 5651	. . . . . 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 6319	. . . . . 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 2252	. . . . 5 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
62	58, 61	elrab2 2966	. . . 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 417	. . 3 ⊢ (𝜑 → (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿))
64		eleq1 2294	. . . 4 ⊢ (𝑟 = (((𝐹‘1Q) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd ‘𝐿) ↔ (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿)))
65	64	rspcev 2911	. . 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 411	. 2 ⊢ (𝜑 → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))
67	42, 66	jca 306	1 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  {crab 2515  ⟨cop 3676   class class class wbr 4093  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028  1^st c1st 6310  2^nd c2nd 6311  Qcnq 7543  1_Qc1q 7544   +_Q cplq 7545   <_Q cltq 7548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616

This theorem is referenced by:  cauappcvgprlemcl  7916