Theorem suplocexprlemex 7985

Description: Lemma for suplocexpr 7988. 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.b	. . 3 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
2		suplocexpr.m	. . . . . 6 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
3		suplocexpr.ub	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
4		suplocexpr.loc	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
5	2, 3, 4	suplocexprlemss 7978	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ P)
6	1	suplocexprlem2b 7977	. . . . 5 ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
7	5, 6	syl 14	. . . 4 ⊢ (𝜑 → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
8	7	opeq2d 3874	. . 3 ⊢ (𝜑 → ⟨∪ (1st “ 𝐴), (2nd ‘𝐵)⟩ = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩)
9	1, 8	eqtr4id 2283	. 2 ⊢ (𝜑 → 𝐵 = ⟨∪ (1st “ 𝐴), (2nd ‘𝐵)⟩)
10		suplocexprlemell 7976	. . . . . . . . 9 ⊢ (𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑡 ∈ 𝐴 𝑠 ∈ (1st ‘𝑡))
11	10	biimpi 120	. . . . . . . 8 ⊢ (𝑠 ∈ ∪ (1st “ 𝐴) → ∃𝑡 ∈ 𝐴 𝑠 ∈ (1st ‘𝑡))
12	11	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) → ∃𝑡 ∈ 𝐴 𝑠 ∈ (1st ‘𝑡))
13	5	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝐴 ⊆ P)
14		simprl 531	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝑡 ∈ 𝐴)
15	13, 14	sseldd 3229	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝑡 ∈ P)
16		prop 7738	. . . . . . . . 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 533	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝑠 ∈ (1st ‘𝑡))
19		elprnql 7744	. . . . . . . 8 ⊢ ((⟨(1st ‘𝑡), (2nd ‘𝑡)⟩ ∈ P ∧ 𝑠 ∈ (1st ‘𝑡)) → 𝑠 ∈ Q)
20	17, 18, 19	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝑠 ∈ Q)
21	12, 20	rexlimddv 2656	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) → 𝑠 ∈ Q)
22	21	ex 115	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ ∪ (1st “ 𝐴) → 𝑠 ∈ Q))
23	22	ssrdv 3234	. . . 4 ⊢ (𝜑 → ∪ (1st “ 𝐴) ⊆ Q)
24		ssrab2 3313	. . . . 5 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ⊆ Q
25	7, 24	eqsstrdi 3280	. . . 4 ⊢ (𝜑 → (2nd ‘𝐵) ⊆ Q)
26	2, 3, 4	suplocexprlemml 7979	. . . . 5 ⊢ (𝜑 → ∃𝑞 ∈ Q 𝑞 ∈ ∪ (1st “ 𝐴))
27	2, 3, 4, 1	suplocexprlemmu 7981	. . . . 5 ⊢ (𝜑 → ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵))
28	26, 27	jca 306	. . . 4 ⊢ (𝜑 → (∃𝑞 ∈ Q 𝑞 ∈ ∪ (1st “ 𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵)))
29	23, 25, 28	jca31 309	. . 3 ⊢ (𝜑 → ((∪ (1st “ 𝐴) ⊆ Q ∧ (2nd ‘𝐵) ⊆ Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ ∪ (1st “ 𝐴) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐵))))
30	2, 3, 4	suplocexprlemrl 7980	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))
31	2, 3, 4, 1	suplocexprlemru 7982	. . . . 5 ⊢ (𝜑 → ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐵) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ (2nd ‘𝐵))))
32	30, 31	jca 306	. . . 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	2, 3, 4, 1	suplocexprlemdisj 7983	. . . 4 ⊢ (𝜑 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))
34	2, 3, 4, 1	suplocexprlemloc 7984	. . . 4 ⊢ (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵))))
35	32, 33, 34	3jca 1204	. . 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 7737	. . 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 417	. 2 ⊢ (𝜑 → ⟨∪ (1st “ 𝐴), (2nd ‘𝐵)⟩ ∈ P)
38	9, 37	eqeltrd 2308	1 ⊢ (𝜑 → 𝐵 ∈ P)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  {crab 2515   ⊆ wss 3201  ⟨cop 3676  ∪ cuni 3898  ∩ cint 3933   class class class wbr 4093   “ cima 4734  ‘cfv 5333  1^st c1st 6310  2^nd c2nd 6311  Qcnq 7543   <_Q cltq 7548  Pcnp 7554  <_P cltp 7558

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-po 4399  df-iso 4400  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-2o 6626  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  df-enq0 7687  df-nq0 7688  df-0nq0 7689  df-plq0 7690  df-mq0 7691  df-inp 7729  df-iltp 7733

This theorem is referenced by:  suplocexprlemub  7986  suplocexpr  7988