Theorem lublecllem 17664

Description: Lemma for lublecl 17665 and lubid 17666. (Contributed by NM, 8-Sep-2018.)

Hypotheses

Ref	Expression
lublecl.b	⊢ 𝐵 = (Base‘𝐾)
lublecl.l	⊢ ≤ = (le‘𝐾)
lublecl.u	⊢ 𝑈 = (lub‘𝐾)
lublecl.k	⊢ (𝜑 → 𝐾 ∈ Poset)
lublecl.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
lublecllem	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)) ↔ 𝑥 = 𝑋))

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧, ≤   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐾,𝑥,𝑧   𝑤,𝑋,𝑥,𝑦,𝑧   𝜑,𝑤,𝑥

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝑈(𝑥,𝑦,𝑧,𝑤)   𝐾(𝑦)

Proof of Theorem lublecllem

Step	Hyp	Ref	Expression
1		breq1 5035	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 ≤ 𝑋 ↔ 𝑧 ≤ 𝑋))
2	1	ralrab 3608	. . 3 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ↔ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
3	1	ralrab 3608	. . . . 5 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 ↔ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤))
4	3	imbi1i 353	. . . 4 ⊢ ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤) ↔ (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
5	4	ralbii 3097	. . 3 ⊢ (∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤) ↔ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
6	2, 5	anbi12i 629	. 2 ⊢ ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)) ↔ (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤)))
7		lublecl.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		lublecl.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Poset)
9		lublecl.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
10		lublecl.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
11	9, 10	posref 17627	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
12	8, 7, 11	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ 𝑋)
13		breq1 5035	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑋 ↔ 𝑋 ≤ 𝑋))
14		breq1 5035	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑥 ↔ 𝑋 ≤ 𝑥))
15	13, 14	imbi12d 348	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → ((𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) ↔ (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑥)))
16	15	rspcva 3539	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥)) → (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑥))
17	12, 16	syl5com 31	. . . . . 6 ⊢ (𝜑 → ((𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥)) → 𝑋 ≤ 𝑥))
18	7, 17	mpand 694	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) → 𝑋 ≤ 𝑥))
19	18	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) → 𝑋 ≤ 𝑥))
20		idd 24	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋))
21	20	rgen 3080	. . . . . 6 ⊢ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋)
22		breq2 5036	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑋 → (𝑧 ≤ 𝑤 ↔ 𝑧 ≤ 𝑋))
23	22	imbi2d 344	. . . . . . . . . 10 ⊢ (𝑤 = 𝑋 → ((𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) ↔ (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋)))
24	23	ralbidv 3126	. . . . . . . . 9 ⊢ (𝑤 = 𝑋 → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) ↔ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋)))
25		breq2 5036	. . . . . . . . 9 ⊢ (𝑤 = 𝑋 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑋))
26	24, 25	imbi12d 348	. . . . . . . 8 ⊢ (𝑤 = 𝑋 → ((∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤) ↔ (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋) → 𝑥 ≤ 𝑋)))
27	26	rspcv 3536	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤) → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋) → 𝑥 ≤ 𝑋)))
28	7, 27	syl 17	. . . . . 6 ⊢ (𝜑 → (∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤) → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋) → 𝑥 ≤ 𝑋)))
29	21, 28	mpii 46	. . . . 5 ⊢ (𝜑 → (∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤) → 𝑥 ≤ 𝑋))
30	29	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤) → 𝑥 ≤ 𝑋))
31	8	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐾 ∈ Poset)
32		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
33	7	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐵)
34	9, 10	posasymb 17628	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ≤ 𝑋 ∧ 𝑋 ≤ 𝑥) ↔ 𝑥 = 𝑋))
35	31, 32, 33, 34	syl3anc 1368	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ≤ 𝑋 ∧ 𝑋 ≤ 𝑥) ↔ 𝑥 = 𝑋))
36	35	biimpd 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ≤ 𝑋 ∧ 𝑋 ≤ 𝑥) → 𝑥 = 𝑋))
37	36	ancomsd 469	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑋 ≤ 𝑥 ∧ 𝑥 ≤ 𝑋) → 𝑥 = 𝑋))
38	19, 30, 37	syl2and 610	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤)) → 𝑥 = 𝑋))
39		breq2 5036	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ 𝑋))
40	39	biimprd 251	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
41	40	ralrimivw 3114	. . . . . 6 ⊢ (𝑥 = 𝑋 → ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
42	41	adantl 485	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
43	7	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑋 ∈ 𝐵)
44		breq1 5035	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑤 ↔ 𝑋 ≤ 𝑤))
45	13, 44	imbi12d 348	. . . . . . . . . 10 ⊢ (𝑧 = 𝑋 → ((𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) ↔ (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤)))
46	45	rspcva 3539	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤)) → (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤))
47		pm5.5 365	. . . . . . . . . . 11 ⊢ (𝑋 ≤ 𝑋 → ((𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤) ↔ 𝑋 ≤ 𝑤))
48	12, 47	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤) ↔ 𝑋 ≤ 𝑤))
49		breq1 5035	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑤 ↔ 𝑋 ≤ 𝑤))
50	49	bicomd 226	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑋 ≤ 𝑤 ↔ 𝑥 ≤ 𝑤))
51	48, 50	sylan9bb 513	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤) ↔ 𝑥 ≤ 𝑤))
52	46, 51	syl5ib 247	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤)) → 𝑥 ≤ 𝑤))
53	43, 52	mpand 694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
54	53	ralrimivw 3114	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
55	54	adantlr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
56	42, 55	jca 515	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 = 𝑋) → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤)))
57	56	ex 416	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 = 𝑋 → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))))
58	38, 57	impbid 215	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤)) ↔ 𝑥 = 𝑋))
59	6, 58	syl5bb 286	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)) ↔ 𝑥 = 𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074   class class class wbr 5032  ‘cfv 6335  Basecbs 16541  lecple 16630  Posetcpo 17616  lubclub 17618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-nul 5176

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-iota 6294  df-fv 6343  df-proset 17604  df-poset 17622

This theorem is referenced by:  lublecl  17665  lubid  17666