Theorem suplocexprlemex 7806

Description: Lemma for suplocexpr 7809. 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 7799	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ P)
6	1	suplocexprlem2b 7798	. . . . 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 3816	. . 3 ⊢ (𝜑 → ⟨∪ (1st “ 𝐴), (2nd ‘𝐵)⟩ = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩)
9	1, 8	eqtr4id 2248	. 2 ⊢ (𝜑 → 𝐵 = ⟨∪ (1st “ 𝐴), (2nd ‘𝐵)⟩)
10		suplocexprlemell 7797	. . . . . . . . 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 529	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝑡 ∈ 𝐴)
15	13, 14	sseldd 3185	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝑡 ∈ P)
16		prop 7559	. . . . . . . . 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 531	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝑠 ∈ (1st ‘𝑡))
19		elprnql 7565	. . . . . . . 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 2619	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) → 𝑠 ∈ Q)
22	21	ex 115	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ ∪ (1st “ 𝐴) → 𝑠 ∈ Q))
23	22	ssrdv 3190	. . . 4 ⊢ (𝜑 → ∪ (1st “ 𝐴) ⊆ Q)
24		ssrab2 3269	. . . . 5 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ⊆ Q
25	7, 24	eqsstrdi 3236	. . . 4 ⊢ (𝜑 → (2nd ‘𝐵) ⊆ Q)
26	2, 3, 4	suplocexprlemml 7800	. . . . 5 ⊢ (𝜑 → ∃𝑞 ∈ Q 𝑞 ∈ ∪ (1st “ 𝐴))
27	2, 3, 4, 1	suplocexprlemmu 7802	. . . . 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 7801	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))
31	2, 3, 4, 1	suplocexprlemru 7803	. . . . 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 7804	. . . 4 ⊢ (𝜑 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))
34	2, 3, 4, 1	suplocexprlemloc 7805	. . . 4 ⊢ (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵))))
35	32, 33, 34	3jca 1179	. . 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 7558	. . 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 2273	1 ⊢ (𝜑 → 𝐵 ∈ P)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 980   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479   ⊆ wss 3157  ⟨cop 3626  ∪ cuni 3840  ∩ cint 3875   class class class wbr 4034   “ cima 4667  ‘cfv 5259  1^st c1st 6205  2^nd c2nd 6206  Qcnq 7364   <_Q cltq 7369  Pcnp 7375  <_P cltp 7379

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iltp 7554

This theorem is referenced by:  suplocexprlemub  7807  suplocexpr  7809