Theorem cauappcvgprlemm 7976

Description: Lemma for cauappcvgpr 7993. 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 5675	. . . . . . 7 ⊢ (𝑝 = 1Q → (𝐹‘𝑝) = (𝐹‘1Q))
2	1	breq2d 4126	. . . . . 6 ⊢ (𝑝 = 1Q → (𝐴 <Q (𝐹‘𝑝) ↔ 𝐴 <Q (𝐹‘1Q)))
3		cauappcvgpr.bnd	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
4		1nq 7697	. . . . . . 7 ⊢ 1Q ∈ Q
5	4	a1i 9	. . . . . 6 ⊢ (𝜑 → 1Q ∈ Q)
6	2, 3, 5	rspcdva 2928	. . . . 5 ⊢ (𝜑 → 𝐴 <Q (𝐹‘1Q))
7		ltrelnq 7696	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
8	7	brel 4807	. . . . . 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 7741	. . . 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 5675	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑠 → (𝐹‘𝑝) = (𝐹‘𝑠))
16	15	breq2d 4126	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑠 → (𝐴 <Q (𝐹‘𝑝) ↔ 𝐴 <Q (𝐹‘𝑠)))
17	16	rspcv 2919	. . . . . . . . . 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 4117	. . . . . . . . 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 6066	. . . . . . . . 9 ⊢ (𝑞 = 𝑠 → (𝑠 +Q 𝑞) = (𝑠 +Q 𝑠))
24		fveq2 5675	. . . . . . . . 9 ⊢ (𝑞 = 𝑠 → (𝐹‘𝑞) = (𝐹‘𝑠))
25	23, 24	breq12d 4127	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → ((𝑠 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑠) <Q (𝐹‘𝑠)))
26	25	rspcev 2923	. . . . . . 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 6065	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → (𝑙 +Q 𝑞) = (𝑠 +Q 𝑞))
29	28	breq1d 4124	. . . . . . . 8 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q 𝑞) <Q (𝐹‘𝑞) ↔ (𝑠 +Q 𝑞) <Q (𝐹‘𝑞)))
30	29	rexbidv 2545	. . . . . . 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 5678	. . . . . . . 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 7694	. . . . . . . . . 10 ⊢ Q ∈ V
34	33	rabex 4261	. . . . . . . . 9 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ V
35	33	rabex 4261	. . . . . . . . 9 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ V
36	34, 35	op1st 6353	. . . . . . . 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 2255	. . . . . . 7 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}
38	30, 37	elrab2 2979	. . . . . 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 2646	. . 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 5818	. . . . 5 ⊢ (𝜑 → (𝐹‘1Q) ∈ Q)
45		addclnq 7706	. . . . 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 7706	. . . 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 7738	. . . . . 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 5675	. . . . . . . 8 ⊢ (𝑞 = 1Q → (𝐹‘𝑞) = (𝐹‘1Q))
52		id 19	. . . . . . . 8 ⊢ (𝑞 = 1Q → 𝑞 = 1Q)
53	51, 52	oveq12d 6076	. . . . . . 7 ⊢ (𝑞 = 1Q → ((𝐹‘𝑞) +Q 𝑞) = ((𝐹‘1Q) +Q 1Q))
54	53	breq1d 4124	. . . . . 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 2923	. . . . 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 4118	. . . . . 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 2545	. . . . 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 5678	. . . . . 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 6354	. . . . . 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 2255	. . . . 5 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}
62	58, 61	elrab2 2979	. . . 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 2297	. . . 4 ⊢ (𝑟 = (((𝐹‘1Q) +Q 1Q) +Q 1Q) → (𝑟 ∈ (2nd ‘𝐿) ↔ (((𝐹‘1Q) +Q 1Q) +Q 1Q) ∈ (2nd ‘𝐿)))
65	64	rspcev 2923	. . 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 2205  ∀wral 2522  ∃wrex 2523  {crab 2526  ⟨cop 3697   class class class wbr 4114  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  1^st c1st 6345  2^nd c2nd 6346  Qcnq 7611  1_Qc1q 7612   +_Q cplq 7613   <_Q cltq 7616

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684

This theorem is referenced by:  cauappcvgprlemcl  7984