Theorem lublecllem 18405

Description: Lemma for lublecl 18406 and lubid 18407. (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 5146	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 ≤ 𝑋 ↔ 𝑧 ≤ 𝑋))
2	1	ralrab 3699	. . 3 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ↔ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
3	1	ralrab 3699	. . . . 5 ⊢ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 ↔ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤))
4	3	imbi1i 349	. . . 4 ⊢ ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤) ↔ (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
5	4	ralbii 3093	. . 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 628	. 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 18364	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
12	8, 7, 11	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ 𝑋)
13		breq1 5146	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑋 ↔ 𝑋 ≤ 𝑋))
14		breq1 5146	. . . . . . . . 9 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑥 ↔ 𝑋 ≤ 𝑥))
15	13, 14	imbi12d 344	. . . . . . . 8 ⊢ (𝑧 = 𝑋 → ((𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) ↔ (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑥)))
16	15	rspcva 3620	. . . . . . 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 695	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥) → 𝑋 ≤ 𝑥))
19	18	adantr 480	. . . 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 3063	. . . . . 6 ⊢ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋)
22		breq2 5147	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑋 → (𝑧 ≤ 𝑤 ↔ 𝑧 ≤ 𝑋))
23	22	imbi2d 340	. . . . . . . . . 10 ⊢ (𝑤 = 𝑋 → ((𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) ↔ (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋)))
24	23	ralbidv 3178	. . . . . . . . 9 ⊢ (𝑤 = 𝑋 → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) ↔ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋)))
25		breq2 5147	. . . . . . . . 9 ⊢ (𝑤 = 𝑋 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑋))
26	24, 25	imbi12d 344	. . . . . . . 8 ⊢ (𝑤 = 𝑋 → ((∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤) ↔ (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑋) → 𝑥 ≤ 𝑋)))
27	26	rspcv 3618	. . . . . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤) → 𝑥 ≤ 𝑋))
31	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐾 ∈ Poset)
32		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
33	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐵)
34	9, 10	posasymb 18365	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ≤ 𝑋 ∧ 𝑋 ≤ 𝑥) ↔ 𝑥 = 𝑋))
35	31, 32, 33, 34	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ≤ 𝑋 ∧ 𝑋 ≤ 𝑥) ↔ 𝑥 = 𝑋))
36	35	biimpd 229	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑥 ≤ 𝑋 ∧ 𝑋 ≤ 𝑥) → 𝑥 = 𝑋))
37	36	ancomsd 465	. . . 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 5147	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ 𝑋))
40	39	biimprd 248	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
41	40	ralrimivw 3150	. . . . . 6 ⊢ (𝑥 = 𝑋 → ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
42	41	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑥))
43	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑋 ∈ 𝐵)
44		breq1 5146	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑤 ↔ 𝑋 ≤ 𝑤))
45	13, 44	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑧 = 𝑋 → ((𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) ↔ (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤)))
46	45	rspcva 3620	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤)) → (𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤))
47		pm5.5 361	. . . . . . . . . . 11 ⊢ (𝑋 ≤ 𝑋 → ((𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤) ↔ 𝑋 ≤ 𝑤))
48	12, 47	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤) ↔ 𝑋 ≤ 𝑤))
49		breq1 5146	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑤 ↔ 𝑋 ≤ 𝑤))
50	49	bicomd 223	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑋 ≤ 𝑤 ↔ 𝑥 ≤ 𝑤))
51	48, 50	sylan9bb 509	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑋 ≤ 𝑋 → 𝑋 ≤ 𝑤) ↔ 𝑥 ≤ 𝑤))
52	46, 51	imbitrid 244	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑋 ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤)) → 𝑥 ≤ 𝑤))
53	43, 52	mpand 695	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
54	53	ralrimivw 3150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
55	54	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 = 𝑋) → ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ 𝐵 (𝑧 ≤ 𝑋 → 𝑧 ≤ 𝑤) → 𝑥 ≤ 𝑤))
56	42, 55	jca 511	. . . 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 412	. . 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 212	. 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 283	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436   class class class wbr 5143  ‘cfv 6561  Basecbs 17247  lecple 17304  Posetcpo 18353  lubclub 18355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-proset 18340  df-poset 18359

This theorem is referenced by:  lublecl  18406  lubid  18407