Theorem suplocexprlemex 7848

Description: Lemma for suplocexpr 7851. 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 7841	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ P)
6	1	suplocexprlem2b 7840	. . . . 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 3829	. . 3 ⊢ (𝜑 → ⟨∪ (1st “ 𝐴), (2nd ‘𝐵)⟩ = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩)
9	1, 8	eqtr4id 2258	. 2 ⊢ (𝜑 → 𝐵 = ⟨∪ (1st “ 𝐴), (2nd ‘𝐵)⟩)
10		suplocexprlemell 7839	. . . . . . . . 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 3196	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) ∧ (𝑡 ∈ 𝐴 ∧ 𝑠 ∈ (1st ‘𝑡))) → 𝑡 ∈ P)
16		prop 7601	. . . . . . . . 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 7607	. . . . . . . 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 2629	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ (1st “ 𝐴)) → 𝑠 ∈ Q)
22	21	ex 115	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ ∪ (1st “ 𝐴) → 𝑠 ∈ Q))
23	22	ssrdv 3201	. . . 4 ⊢ (𝜑 → ∪ (1st “ 𝐴) ⊆ Q)
24		ssrab2 3280	. . . . 5 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ⊆ Q
25	7, 24	eqsstrdi 3247	. . . 4 ⊢ (𝜑 → (2nd ‘𝐵) ⊆ Q)
26	2, 3, 4	suplocexprlemml 7842	. . . . 5 ⊢ (𝜑 → ∃𝑞 ∈ Q 𝑞 ∈ ∪ (1st “ 𝐴))
27	2, 3, 4, 1	suplocexprlemmu 7844	. . . . 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 7843	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))
31	2, 3, 4, 1	suplocexprlemru 7845	. . . . 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 7846	. . . 4 ⊢ (𝜑 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))
34	2, 3, 4, 1	suplocexprlemloc 7847	. . . 4 ⊢ (𝜑 → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <Q 𝑟 → (𝑞 ∈ ∪ (1st “ 𝐴) ∨ 𝑟 ∈ (2nd ‘𝐵))))
35	32, 33, 34	3jca 1180	. . 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 7600	. . 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 2283	1 ⊢ (𝜑 → 𝐵 ∈ P)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710   ∧ w3a 981   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  {crab 2489   ⊆ wss 3168  ⟨cop 3638  ∪ cuni 3853  ∩ cint 3888   class class class wbr 4048   “ cima 4683  ‘cfv 5277  1^st c1st 6234  2^nd c2nd 6235  Qcnq 7406   <_Q cltq 7411  Pcnp 7417  <_P cltp 7421

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-eprel 4341  df-id 4345  df-po 4348  df-iso 4349  df-iord 4418  df-on 4420  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-recs 6401  df-irdg 6466  df-1o 6512  df-2o 6513  df-oadd 6516  df-omul 6517  df-er 6630  df-ec 6632  df-qs 6636  df-ni 7430  df-pli 7431  df-mi 7432  df-lti 7433  df-plpq 7470  df-mpq 7471  df-enq 7473  df-nqqs 7474  df-plqqs 7475  df-mqqs 7476  df-1nqqs 7477  df-rq 7478  df-ltnqqs 7479  df-enq0 7550  df-nq0 7551  df-0nq0 7552  df-plq0 7553  df-mq0 7554  df-inp 7592  df-iltp 7596

This theorem is referenced by:  suplocexprlemub  7849  suplocexpr  7851