Theorem lublecllem 16988

Description: Lemma for lublecl 16989 and lubid 16990. (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 4656	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 ≤ 𝑋 ↔ 𝑧 ≤ 𝑋))
2	1	ralrab 3368	. . 3 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ↔ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
3	1	ralrab 3368	. . . . 5 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 ↔ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤))
4	3	imbi1i 339	. . . 4 ⊢ ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤) ↔ (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
5	4	ralbii 2980	. . 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 733	. 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 16951	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
12	8, 7, 11	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ 𝑋)
13		breq1 4656	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑋 ↔ 𝑋 ≤ 𝑋))
14		breq1 4656	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑥 ↔ 𝑋 ≤ 𝑥))
15	13, 14	imbi12d 334	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → ((𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) ↔ (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑥)))
16	15	rspcva 3307	. . . . . . 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 711	. . . . 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 2922	. . . . . 6 ⊢ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋)
22		breq2 4657	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑋 → (𝑧 ≤ 𝑤 ↔ 𝑧 ≤ 𝑋))
23	22	imbi2d 330	. . . . . . . . . 10 ⊢ (𝑤 = 𝑋 → ((𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) ↔ (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋)))
24	23	ralbidv 2986	. . . . . . . . 9 ⊢ (𝑤 = 𝑋 → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) ↔ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋)))
25		breq2 4657	. . . . . . . . 9 ⊢ (𝑤 = 𝑋 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑋))
26	24, 25	imbi12d 334	. . . . . . . 8 ⊢ (𝑤 = 𝑋 → ((∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤) ↔ (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋) → 𝑥 ≤ 𝑋)))
27	26	rspcv 3305	. . . . . . 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 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
33	7	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐵)
34	9, 10	posasymb 16952	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ≤ 𝑋 ∧ 𝑋 ≤ 𝑥) ↔ 𝑥 = 𝑋))
35	31, 32, 33, 34	syl3anc 1326	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ≤ 𝑋 ∧ 𝑋 ≤ 𝑥) ↔ 𝑥 = 𝑋))
36	35	biimpd 219	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ≤ 𝑋 ∧ 𝑋 ≤ 𝑥) → 𝑥 = 𝑋))
37	36	ancomsd 470	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑋 ≤ 𝑥 ∧ 𝑥 ≤ 𝑋) → 𝑥 = 𝑋))
38	19, 30, 37	syl2and 500	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤)) → 𝑥 = 𝑋))
39		breq2 4657	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ 𝑋))
40	39	biimprd 238	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
41	40	ralrimivw 2967	. . . . . 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 4656	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑤 ↔ 𝑋 ≤ 𝑤))
45	13, 44	imbi12d 334	. . . . . . . . . 10 ⊢ (𝑧 = 𝑋 → ((𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) ↔ (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤)))
46	45	rspcva 3307	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤)) → (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤))
47		pm5.5 351	. . . . . . . . . . 11 ⊢ (𝑋 ≤ 𝑋 → ((𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤) ↔ 𝑋 ≤ 𝑤))
48	12, 47	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤) ↔ 𝑋 ≤ 𝑤))
49		breq1 4656	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑤 ↔ 𝑋 ≤ 𝑤))
50	49	bicomd 213	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑋 ≤ 𝑤 ↔ 𝑥 ≤ 𝑤))
51	48, 50	sylan9bb 736	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤) ↔ 𝑥 ≤ 𝑤))
52	46, 51	syl5ib 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤)) → 𝑥 ≤ 𝑤))
53	43, 52	mpand 711	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
54	53	ralrimivw 2967	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
55	54	adantlr 751	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
56	42, 55	jca 554	. . . 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 450	. . 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 202	. 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 272	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   class class class wbr 4653  ‘cfv 5888  Basecbs 15857  lecple 15948  Posetcpo 16940  lubclub 16942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-preset 16928  df-poset 16946

This theorem is referenced by:  lublecl  16989  lubid  16990