Theorem suplocexprlemex 7530

Description: Lemma for suplocexpr 7533. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
suplocexpr.b	⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩

Assertion

Ref	Expression
suplocexprlemex	⊢ (𝜑 → 𝐵 ∈ P)

Distinct variable groups:   𝑢,𝐴,𝑤,𝑧   𝑥,𝐴,𝑢,𝑦,𝑧   𝑤,𝐵   𝜑,𝑢,𝑤,𝑧   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑢)

Proof of Theorem suplocexprlemex

Dummy variables 𝑞 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suplocexpr.m	. . . . . 6 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
2		suplocexpr.ub	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
3		suplocexpr.loc	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
4	1, 2, 3	suplocexprlemss 7523	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ P)
5		suplocexpr.b	. . . . . 6 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
6	5	suplocexprlem2b 7522	. . . . 5 ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
7	4, 6	syl 14	. . . 4 ⊢ (𝜑 → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
8	7	opeq2d 3712	. . 3 ⊢ (𝜑 → ⟨∪ (1st “ 𝐴), (2nd ‘𝐵)⟩ = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩)
9	8, 5	syl6reqr 2191	. 2 ⊢ (𝜑 → 𝐵 = ⟨∪ (1st “ 𝐴), (2nd ‘𝐵)⟩)
10		suplocexprlemell 7521	. . . . . . . . 9 ⊢ (𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑡 ∈ 𝐴 𝑠 ∈ (1st ‘𝑡))
11	10	biimpi 119	. . . . . . . 8 ⊢ (𝑠 ∈ ∪ (1st “ 𝐴) → ∃𝑡 ∈ 𝐴 𝑠 ∈ (1st ‘𝑡))
12	11	adantl 275	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) → ∃𝑡 ∈ 𝐴 𝑠 ∈ (1st ‘𝑡))
13	4	ad2antrr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝐴 ⊆ P)
14		simprl 520	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝑡 ∈ 𝐴)
15	13, 14	sseldd 3098	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝑡 ∈ P)
16		prop 7283	. . . . . . . . 9 ⊢ (𝑡 ∈ P → ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P)
17	15, 16	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → ⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P)
18		simprr 521	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝑠 ∈ (1st ‘𝑡))
19		elprnql 7289	. . . . . . . 8 ⊢ ((⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P ∧ 𝑠 ∈ (1st ‘𝑡)) → 𝑠 ∈ Q)
20	17, 18, 19	syl2anc 408	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝑠 ∈ Q)
21	12, 20	rexlimddv 2554	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) → 𝑠 ∈ Q)
22	21	ex 114	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ ∪ (1st “ 𝐴) → 𝑠 ∈ Q))
23	22	ssrdv 3103	. . . 4 ⊢ (𝜑 → ∪ (1st “ 𝐴) ⊆ Q)
24		ssrab2 3182	. . . . 5 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ⊆ Q
25	7, 24	eqsstrdi 3149	. . . 4 ⊢ (𝜑 → (2nd ‘𝐵) ⊆ Q)
26	1, 2, 3	suplocexprlemml 7524	. . . . 5 ⊢ (𝜑 → ∃𝑞 ∈ Q 𝑞 ∈ ∪ (1st “ 𝐴))
27	1, 2, 3, 5	suplocexprlemmu 7526	. . . . 5 ⊢ (𝜑 → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵))
28	26, 27	jca 304	. . . 4 ⊢ (𝜑 → (∃𝑞 ∈ Q 𝑞 ∈ ∪ (1st “ 𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵)))
29	23, 25, 28	jca31 307	. . 3 ⊢ (𝜑 → ((∪ (1st “ 𝐴) ⊆ Q ∧ (2nd ‘𝐵) ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ ∪ (1st “ 𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵))))
30	1, 2, 3	suplocexprlemrl 7525	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))
31	1, 2, 3, 5	suplocexprlemru 7527	. . . . 5 ⊢ (𝜑 → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))
32	30, 31	jca 304	. . . 4 ⊢ (𝜑 → (∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))))
33	1, 2, 3, 5	suplocexprlemdisj 7528	. . . 4 ⊢ (𝜑 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))
34	1, 2, 3, 5	suplocexprlemloc 7529	. . . 4 ⊢ (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵))))
35	32, 33, 34	3jca 1161	. . 3 ⊢ (𝜑 → ((∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵)))))
36		elinp 7282	. . 3 ⊢ (⟨∪ (1st “ 𝐴), (2nd ‘𝐵)⟩ ∈ P ↔ (((∪ (1st “ 𝐴) ⊆ Q ∧ (2nd ‘𝐵) ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ ∪ (1st “ 𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵)))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵))))))
37	29, 35, 36	sylanbrc 413	. 2 ⊢ (𝜑 → ⟨∪ (1st “ 𝐴), (2nd ‘𝐵)⟩ ∈ P)
38	9, 37	eqeltrd 2216	1 ⊢ (𝜑 → 𝐵 ∈ P)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  {crab 2420   ⊆ wss 3071  ⟨cop 3530  ∪ cuni 3736  ∩ cint 3771   class class class wbr 3929   “ cima 4542  ‘cfv 5123  1^st c1st 6036  2^nd c2nd 6037  Qcnq 7088   <_Q cltq 7093  Pcnp 7099  <_P cltp 7103

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-po 4218  df-iso 4219  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-2o 6314  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  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iltp 7278

This theorem is referenced by:  suplocexprlemub  7531  suplocexpr  7533