Theorem lublecllem 18078

Description: Lemma for lublecl 18079 and lubid 18080. (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 5077	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 ≤ 𝑋 ↔ 𝑧 ≤ 𝑋))
2	1	ralrab 3630	. . 3 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ↔ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
3	1	ralrab 3630	. . . . 5 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 ↔ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤))
4	3	imbi1i 350	. . . 4 ⊢ ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤) ↔ (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
5	4	ralbii 3092	. . 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 627	. 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 18036	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
12	8, 7, 11	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ 𝑋)
13		breq1 5077	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑋 ↔ 𝑋 ≤ 𝑋))
14		breq1 5077	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑥 ↔ 𝑋 ≤ 𝑥))
15	13, 14	imbi12d 345	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → ((𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) ↔ (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑥)))
16	15	rspcva 3559	. . . . . . 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 692	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) → 𝑋 ≤ 𝑥))
19	18	adantr 481	. . . 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 3074	. . . . . 6 ⊢ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋)
22		breq2 5078	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑋 → (𝑧 ≤ 𝑤 ↔ 𝑧 ≤ 𝑋))
23	22	imbi2d 341	. . . . . . . . . 10 ⊢ (𝑤 = 𝑋 → ((𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) ↔ (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋)))
24	23	ralbidv 3112	. . . . . . . . 9 ⊢ (𝑤 = 𝑋 → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) ↔ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋)))
25		breq2 5078	. . . . . . . . 9 ⊢ (𝑤 = 𝑋 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑋))
26	24, 25	imbi12d 345	. . . . . . . 8 ⊢ (𝑤 = 𝑋 → ((∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤) ↔ (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋) → 𝑥 ≤ 𝑋)))
27	26	rspcv 3557	. . . . . . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤) → 𝑥 ≤ 𝑋))
31	8	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐾 ∈ Poset)
32		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
33	7	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐵)
34	9, 10	posasymb 18037	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ≤ 𝑋 ∧ 𝑋 ≤ 𝑥) ↔ 𝑥 = 𝑋))
35	31, 32, 33, 34	syl3anc 1370	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ≤ 𝑋 ∧ 𝑋 ≤ 𝑥) ↔ 𝑥 = 𝑋))
36	35	biimpd 228	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ≤ 𝑋 ∧ 𝑋 ≤ 𝑥) → 𝑥 = 𝑋))
37	36	ancomsd 466	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑋 ≤ 𝑥 ∧ 𝑥 ≤ 𝑋) → 𝑥 = 𝑋))
38	19, 30, 37	syl2and 608	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤)) → 𝑥 = 𝑋))
39		breq2 5078	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ 𝑋))
40	39	biimprd 247	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
41	40	ralrimivw 3104	. . . . . 6 ⊢ (𝑥 = 𝑋 → ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
42	41	adantl 482	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
43	7	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑋 ∈ 𝐵)
44		breq1 5077	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑤 ↔ 𝑋 ≤ 𝑤))
45	13, 44	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑧 = 𝑋 → ((𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) ↔ (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤)))
46	45	rspcva 3559	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤)) → (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤))
47		pm5.5 362	. . . . . . . . . . 11 ⊢ (𝑋 ≤ 𝑋 → ((𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤) ↔ 𝑋 ≤ 𝑤))
48	12, 47	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤) ↔ 𝑋 ≤ 𝑤))
49		breq1 5077	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑤 ↔ 𝑋 ≤ 𝑤))
50	49	bicomd 222	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑋 ≤ 𝑤 ↔ 𝑥 ≤ 𝑤))
51	48, 50	sylan9bb 510	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤) ↔ 𝑥 ≤ 𝑤))
52	46, 51	syl5ib 243	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤)) → 𝑥 ≤ 𝑤))
53	43, 52	mpand 692	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
54	53	ralrimivw 3104	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
55	54	adantlr 712	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
56	42, 55	jca 512	. . . 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 413	. . 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 211	. 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	bitrid 282	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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   class class class wbr 5074  ‘cfv 6433  Basecbs 16912  lecple 16969  Posetcpo 18025  lubclub 18027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-proset 18013  df-poset 18031

This theorem is referenced by:  lublecl  18079  lubid  18080