Theorem suplocexprlemrl 7679

Description: Lemma for suplocexpr 7687. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-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)))

Assertion

Ref	Expression
suplocexprlemrl	⊢ (𝜑 → ∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))

Distinct variable groups:   𝐴,𝑟   𝑥,𝐴,𝑦   𝜑,𝑞,𝑟   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧,𝑞)

Proof of Theorem suplocexprlemrl

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suplocexprlemell 7675	. . . . . . 7 ⊢ (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠))
2	1	biimpi 119	. . . . . 6 ⊢ (𝑞 ∈ ∪ (1st “ 𝐴) → ∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠))
3	2	adantl 275	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) → ∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠))
4		suplocexpr.m	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
5		suplocexpr.ub	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
6		suplocexpr.loc	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
7	4, 5, 6	suplocexprlemss 7677	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ P)
8	7	ad3antrrr 489	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝐴 ⊆ P)
9		simprl 526	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑠 ∈ 𝐴)
10	8, 9	sseldd 3148	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑠 ∈ P)
11		prop 7437	. . . . . . . 8 ⊢ (𝑠 ∈ P → ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P)
12	10, 11	syl 14	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P)
13		simprr 527	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑞 ∈ (1st ‘𝑠))
14		prnmaxl 7450	. . . . . . 7 ⊢ ((⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝑠)) → ∃𝑟 ∈ (1st ‘𝑠)𝑞 <Q 𝑟)
15	12, 13, 14	syl2anc 409	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ∃𝑟 ∈ (1st ‘𝑠)𝑞 <Q 𝑟)
16		ltrelnq 7327	. . . . . . . . 9 ⊢ <Q ⊆ (Q × Q)
17	16	brel 4663	. . . . . . . 8 ⊢ (𝑞 <Q 𝑟 → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
18	17	simprd 113	. . . . . . 7 ⊢ (𝑞 <Q 𝑟 → 𝑟 ∈ Q)
19	18	ad2antll 488	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑟 ∈ Q)
20		simprr 527	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑞 <Q 𝑟)
21		simplrl 530	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑠 ∈ 𝐴)
22		simprl 526	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑟 ∈ (1st ‘𝑠))
23		rspe 2519	. . . . . . . . 9 ⊢ ((𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠)) → ∃𝑠 ∈ 𝐴 𝑟 ∈ (1st ‘𝑠))
24	21, 22, 23	syl2anc 409	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → ∃𝑠 ∈ 𝐴 𝑟 ∈ (1st ‘𝑠))
25		suplocexprlemell 7675	. . . . . . . 8 ⊢ (𝑟 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑠 ∈ 𝐴 𝑟 ∈ (1st ‘𝑠))
26	24, 25	sylibr 133	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑟 ∈ ∪ (1st “ 𝐴))
27	20, 26	jca 304	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴)))
28	15, 19, 27	reximssdv 2574	. . . . 5 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴)))
29	3, 28	rexlimddv 2592	. . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴)))
30	29	ex 114	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ Q) → (𝑞 ∈ ∪ (1st “ 𝐴) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))
31		simprr 527	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) → 𝑟 ∈ ∪ (1st “ 𝐴))
32	31, 25	sylib 121	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) → ∃𝑠 ∈ 𝐴 𝑟 ∈ (1st ‘𝑠))
33		simprl 526	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝑠 ∈ 𝐴)
34		simplrl 530	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝑞 <Q 𝑟)
35	7	ad3antrrr 489	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝐴 ⊆ P)
36	35, 33	sseldd 3148	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝑠 ∈ P)
37	36, 11	syl 14	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P)
38		simprr 527	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝑟 ∈ (1st ‘𝑠))
39		prcdnql 7446	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P ∧ 𝑟 ∈ (1st ‘𝑠)) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑠)))
40	37, 38, 39	syl2anc 409	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑠)))
41	34, 40	mpd 13	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝑞 ∈ (1st ‘𝑠))
42		19.8a 1583	. . . . . . . . 9 ⊢ ((𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠)) → ∃𝑠(𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠)))
43	33, 41, 42	syl2anc 409	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → ∃𝑠(𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠)))
44		df-rex 2454	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠) ↔ ∃𝑠(𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠)))
45	43, 44	sylibr 133	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → ∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠))
46	45, 1	sylibr 133	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝑞 ∈ ∪ (1st “ 𝐴))
47	32, 46	rexlimddv 2592	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) → 𝑞 ∈ ∪ (1st “ 𝐴))
48	47	ex 114	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ Q) → ((𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴)) → 𝑞 ∈ ∪ (1st “ 𝐴)))
49	48	rexlimdvw 2591	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴)) → 𝑞 ∈ ∪ (1st “ 𝐴)))
50	30, 49	impbid 128	. 2 ⊢ ((𝜑 ∧ 𝑞 ∈ Q) → (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))
51	50	ralrimiva 2543	1 ⊢ (𝜑 → ∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  ∃wex 1485   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449   ⊆ wss 3121  ⟨cop 3586  ∪ cuni 3796   class class class wbr 3989   “ cima 4614  ‘cfv 5198  1^st c1st 6117  2^nd c2nd 6118  Qcnq 7242   <_Q cltq 7247  Pcnp 7253  <_P cltp 7257

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-qs 6519  df-ni 7266  df-nqqs 7310  df-ltnqqs 7315  df-inp 7428  df-iltp 7432

This theorem is referenced by:  suplocexprlemex  7684