Theorem suplocexprlemrl 7718

Description: Lemma for suplocexpr 7726. 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 7714	. . . . . . 7 ⊢ (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠))
2	1	biimpi 120	. . . . . 6 ⊢ (𝑞 ∈ ∪ (1st “ 𝐴) → ∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠))
3	2	adantl 277	. . . . 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 7716	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ P)
8	7	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝐴 ⊆ P)
9		simprl 529	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑠 ∈ 𝐴)
10	8, 9	sseldd 3158	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑠 ∈ P)
11		prop 7476	. . . . . . . 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 531	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑞 ∈ (1st ‘𝑠))
14		prnmaxl 7489	. . . . . . 7 ⊢ ((⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P ∧ 𝑞 ∈ (1st ‘𝑠)) → ∃𝑟 ∈ (1st ‘𝑠)𝑞 <Q 𝑟)
15	12, 13, 14	syl2anc 411	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ∃𝑟 ∈ (1st ‘𝑠)𝑞 <Q 𝑟)
16		ltrelnq 7366	. . . . . . . . 9 ⊢ <Q ⊆ (Q × Q)
17	16	brel 4680	. . . . . . . 8 ⊢ (𝑞 <Q 𝑟 → (𝑞 ∈ Q ∧ 𝑟 ∈ Q))
18	17	simprd 114	. . . . . . 7 ⊢ (𝑞 <Q 𝑟 → 𝑟 ∈ Q)
19	18	ad2antll 491	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑟 ∈ Q)
20		simprr 531	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑞 <Q 𝑟)
21		simplrl 535	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑠 ∈ 𝐴)
22		simprl 529	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑟 ∈ (1st ‘𝑠))
23		rspe 2526	. . . . . . . . 9 ⊢ ((𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠)) → ∃𝑠 ∈ 𝐴 𝑟 ∈ (1st ‘𝑠))
24	21, 22, 23	syl2anc 411	. . . . . . . 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 7714	. . . . . . . 8 ⊢ (𝑟 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑠 ∈ 𝐴 𝑟 ∈ (1st ‘𝑠))
26	24, 25	sylibr 134	. . . . . . 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 306	. . . . . 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 2581	. . . . 5 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴)))
29	3, 28	rexlimddv 2599	. . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴)))
30	29	ex 115	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ Q) → (𝑞 ∈ ∪ (1st “ 𝐴) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))
31		simprr 531	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) → 𝑟 ∈ ∪ (1st “ 𝐴))
32	31, 25	sylib 122	. . . . . 6 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) → ∃𝑠 ∈ 𝐴 𝑟 ∈ (1st ‘𝑠))
33		simprl 529	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝑠 ∈ 𝐴)
34		simplrl 535	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝑞 <Q 𝑟)
35	7	ad3antrrr 492	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝐴 ⊆ P)
36	35, 33	sseldd 3158	. . . . . . . . . . . 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 531	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝑟 ∈ (1st ‘𝑠))
39		prcdnql 7485	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P ∧ 𝑟 ∈ (1st ‘𝑠)) → (𝑞 <Q 𝑟 → 𝑞 ∈ (1st ‘𝑠)))
40	37, 38, 39	syl2anc 411	. . . . . . . . . 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 1590	. . . . . . . . 9 ⊢ ((𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠)) → ∃𝑠(𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠)))
43	33, 41, 42	syl2anc 411	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → ∃𝑠(𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠)))
44		df-rex 2461	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠) ↔ ∃𝑠(𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠)))
45	43, 44	sylibr 134	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → ∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠))
46	45, 1	sylibr 134	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝑞 ∈ ∪ (1st “ 𝐴))
47	32, 46	rexlimddv 2599	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) → 𝑞 ∈ ∪ (1st “ 𝐴))
48	47	ex 115	. . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ Q) → ((𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴)) → 𝑞 ∈ ∪ (1st “ 𝐴)))
49	48	rexlimdvw 2598	. . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴)) → 𝑞 ∈ ∪ (1st “ 𝐴)))
50	30, 49	impbid 129	. 2 ⊢ ((𝜑 ∧ 𝑞 ∈ Q) → (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))
51	50	ralrimiva 2550	1 ⊢ (𝜑 → ∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456   ⊆ wss 3131  ⟨cop 3597  ∪ cuni 3811   class class class wbr 4005   “ cima 4631  ‘cfv 5218  1^st c1st 6141  2^nd c2nd 6142  Qcnq 7281   <_Q cltq 7286  Pcnp 7292  <_P cltp 7296

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-qs 6543  df-ni 7305  df-nqqs 7349  df-ltnqqs 7354  df-inp 7467  df-iltp 7471

This theorem is referenced by:  suplocexprlemex  7723