Theorem suplocexprlemrl 7980

Description: Lemma for suplocexpr 7988. 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 7976	. . . . . . 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 7978	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ P)
8	7	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝐴 ⊆ P)
9		simprl 531	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑠 ∈ 𝐴)
10	8, 9	sseldd 3229	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑠 ∈ P)
11		prop 7738	. . . . . . . 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 533	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑞 ∈ (1st ‘𝑠))
14		prnmaxl 7751	. . . . . . 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 7628	. . . . . . . . 9 ⊢ <Q ⊆ (Q × Q)
17	16	brel 4784	. . . . . . . 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 533	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑞 <Q 𝑟)
21		simplrl 537	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑠 ∈ 𝐴)
22		simprl 531	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑟 ∈ (1st ‘𝑠) ∧ 𝑞 <Q 𝑟)) → 𝑟 ∈ (1st ‘𝑠))
23		rspe 2582	. . . . . . . . 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 7976	. . . . . . . 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 2637	. . . . 5 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ 𝑞 ∈ ∪ (1st “ 𝐴)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴)))
29	3, 28	rexlimddv 2656	. . . 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 533	. . . . . . 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 531	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝑠 ∈ 𝐴)
34		simplrl 537	. . . . . . . . . 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 3229	. . . . . . . . . . . 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 533	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑟 ∈ (1st ‘𝑠))) → 𝑟 ∈ (1st ‘𝑠))
39		prcdnql 7747	. . . . . . . . . . 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 1639	. . . . . . . . 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 2517	. . . . . . . 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 2656	. . . . 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 2655	. . 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 2606	1 ⊢ (𝜑 → ∀𝑞 ∈ Q (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ ∪ (1st “ 𝐴))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  ∃wex 1541   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512   ⊆ wss 3201  ⟨cop 3676  ∪ cuni 3898   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-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  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-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-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-id 4396  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-1st 6312  df-2nd 6313  df-qs 6751  df-ni 7567  df-nqqs 7611  df-ltnqqs 7616  df-inp 7729  df-iltp 7733

This theorem is referenced by:  suplocexprlemex  7985