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Theorem lublecllem 17432
Description: Lemma for lublecl 17433 and lubid 17434. (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 (𝜑𝐾 ∈ Poset)
lublecl.x (𝜑𝑋𝐵)
Assertion
Ref Expression
lublecllem ((𝜑𝑥𝐵) → ((∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 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
StepHypRef Expression
1 breq1 4969 . . . 4 (𝑦 = 𝑧 → (𝑦 𝑋𝑧 𝑋))
21ralrab 3624 . . 3 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ↔ ∀𝑧𝐵 (𝑧 𝑋𝑧 𝑥))
31ralrab 3624 . . . . 5 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤 ↔ ∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤))
43imbi1i 351 . . . 4 ((∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤) ↔ (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤))
54ralbii 3132 . . 3 (∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤) ↔ ∀𝑤𝐵 (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤))
62, 5anbi12i 626 . 2 ((∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤)) ↔ (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑥) ∧ ∀𝑤𝐵 (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤)))
7 lublecl.x . . . . . 6 (𝜑𝑋𝐵)
8 lublecl.k . . . . . . . 8 (𝜑𝐾 ∈ Poset)
9 lublecl.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
10 lublecl.l . . . . . . . . 9 = (le‘𝐾)
119, 10posref 17395 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋 𝑋)
128, 7, 11syl2anc 584 . . . . . . 7 (𝜑𝑋 𝑋)
13 breq1 4969 . . . . . . . . 9 (𝑧 = 𝑋 → (𝑧 𝑋𝑋 𝑋))
14 breq1 4969 . . . . . . . . 9 (𝑧 = 𝑋 → (𝑧 𝑥𝑋 𝑥))
1513, 14imbi12d 346 . . . . . . . 8 (𝑧 = 𝑋 → ((𝑧 𝑋𝑧 𝑥) ↔ (𝑋 𝑋𝑋 𝑥)))
1615rspcva 3557 . . . . . . 7 ((𝑋𝐵 ∧ ∀𝑧𝐵 (𝑧 𝑋𝑧 𝑥)) → (𝑋 𝑋𝑋 𝑥))
1712, 16syl5com 31 . . . . . 6 (𝜑 → ((𝑋𝐵 ∧ ∀𝑧𝐵 (𝑧 𝑋𝑧 𝑥)) → 𝑋 𝑥))
187, 17mpand 691 . . . . 5 (𝜑 → (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑥) → 𝑋 𝑥))
1918adantr 481 . . . 4 ((𝜑𝑥𝐵) → (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑥) → 𝑋 𝑥))
20 idd 24 . . . . . . 7 (𝑧𝐵 → (𝑧 𝑋𝑧 𝑋))
2120rgen 3115 . . . . . 6 𝑧𝐵 (𝑧 𝑋𝑧 𝑋)
22 breq2 4970 . . . . . . . . . . 11 (𝑤 = 𝑋 → (𝑧 𝑤𝑧 𝑋))
2322imbi2d 342 . . . . . . . . . 10 (𝑤 = 𝑋 → ((𝑧 𝑋𝑧 𝑤) ↔ (𝑧 𝑋𝑧 𝑋)))
2423ralbidv 3164 . . . . . . . . 9 (𝑤 = 𝑋 → (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) ↔ ∀𝑧𝐵 (𝑧 𝑋𝑧 𝑋)))
25 breq2 4970 . . . . . . . . 9 (𝑤 = 𝑋 → (𝑥 𝑤𝑥 𝑋))
2624, 25imbi12d 346 . . . . . . . 8 (𝑤 = 𝑋 → ((∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤) ↔ (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑋) → 𝑥 𝑋)))
2726rspcv 3555 . . . . . . 7 (𝑋𝐵 → (∀𝑤𝐵 (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤) → (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑋) → 𝑥 𝑋)))
287, 27syl 17 . . . . . 6 (𝜑 → (∀𝑤𝐵 (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤) → (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑋) → 𝑥 𝑋)))
2921, 28mpii 46 . . . . 5 (𝜑 → (∀𝑤𝐵 (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤) → 𝑥 𝑋))
3029adantr 481 . . . 4 ((𝜑𝑥𝐵) → (∀𝑤𝐵 (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤) → 𝑥 𝑋))
318adantr 481 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐾 ∈ Poset)
32 simpr 485 . . . . . . 7 ((𝜑𝑥𝐵) → 𝑥𝐵)
337adantr 481 . . . . . . 7 ((𝜑𝑥𝐵) → 𝑋𝐵)
349, 10posasymb 17396 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑋𝐵) → ((𝑥 𝑋𝑋 𝑥) ↔ 𝑥 = 𝑋))
3531, 32, 33, 34syl3anc 1364 . . . . . 6 ((𝜑𝑥𝐵) → ((𝑥 𝑋𝑋 𝑥) ↔ 𝑥 = 𝑋))
3635biimpd 230 . . . . 5 ((𝜑𝑥𝐵) → ((𝑥 𝑋𝑋 𝑥) → 𝑥 = 𝑋))
3736ancomsd 466 . . . 4 ((𝜑𝑥𝐵) → ((𝑋 𝑥𝑥 𝑋) → 𝑥 = 𝑋))
3819, 30, 37syl2and 607 . . 3 ((𝜑𝑥𝐵) → ((∀𝑧𝐵 (𝑧 𝑋𝑧 𝑥) ∧ ∀𝑤𝐵 (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤)) → 𝑥 = 𝑋))
39 breq2 4970 . . . . . . . 8 (𝑥 = 𝑋 → (𝑧 𝑥𝑧 𝑋))
4039biimprd 249 . . . . . . 7 (𝑥 = 𝑋 → (𝑧 𝑋𝑧 𝑥))
4140ralrimivw 3150 . . . . . 6 (𝑥 = 𝑋 → ∀𝑧𝐵 (𝑧 𝑋𝑧 𝑥))
4241adantl 482 . . . . 5 (((𝜑𝑥𝐵) ∧ 𝑥 = 𝑋) → ∀𝑧𝐵 (𝑧 𝑋𝑧 𝑥))
437adantr 481 . . . . . . . 8 ((𝜑𝑥 = 𝑋) → 𝑋𝐵)
44 breq1 4969 . . . . . . . . . . 11 (𝑧 = 𝑋 → (𝑧 𝑤𝑋 𝑤))
4513, 44imbi12d 346 . . . . . . . . . 10 (𝑧 = 𝑋 → ((𝑧 𝑋𝑧 𝑤) ↔ (𝑋 𝑋𝑋 𝑤)))
4645rspcva 3557 . . . . . . . . 9 ((𝑋𝐵 ∧ ∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤)) → (𝑋 𝑋𝑋 𝑤))
47 pm5.5 363 . . . . . . . . . . 11 (𝑋 𝑋 → ((𝑋 𝑋𝑋 𝑤) ↔ 𝑋 𝑤))
4812, 47syl 17 . . . . . . . . . 10 (𝜑 → ((𝑋 𝑋𝑋 𝑤) ↔ 𝑋 𝑤))
49 breq1 4969 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝑥 𝑤𝑋 𝑤))
5049bicomd 224 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑋 𝑤𝑥 𝑤))
5148, 50sylan9bb 510 . . . . . . . . 9 ((𝜑𝑥 = 𝑋) → ((𝑋 𝑋𝑋 𝑤) ↔ 𝑥 𝑤))
5246, 51syl5ib 245 . . . . . . . 8 ((𝜑𝑥 = 𝑋) → ((𝑋𝐵 ∧ ∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤)) → 𝑥 𝑤))
5343, 52mpand 691 . . . . . . 7 ((𝜑𝑥 = 𝑋) → (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤))
5453ralrimivw 3150 . . . . . 6 ((𝜑𝑥 = 𝑋) → ∀𝑤𝐵 (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤))
5554adantlr 711 . . . . 5 (((𝜑𝑥𝐵) ∧ 𝑥 = 𝑋) → ∀𝑤𝐵 (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤))
5642, 55jca 512 . . . 4 (((𝜑𝑥𝐵) ∧ 𝑥 = 𝑋) → (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑥) ∧ ∀𝑤𝐵 (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤)))
5756ex 413 . . 3 ((𝜑𝑥𝐵) → (𝑥 = 𝑋 → (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑥) ∧ ∀𝑤𝐵 (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤))))
5838, 57impbid 213 . 2 ((𝜑𝑥𝐵) → ((∀𝑧𝐵 (𝑧 𝑋𝑧 𝑥) ∧ ∀𝑤𝐵 (∀𝑧𝐵 (𝑧 𝑋𝑧 𝑤) → 𝑥 𝑤)) ↔ 𝑥 = 𝑋))
596, 58syl5bb 284 1 ((𝜑𝑥𝐵) → ((∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑥 ∧ ∀𝑤𝐵 (∀𝑧 ∈ {𝑦𝐵𝑦 𝑋}𝑧 𝑤𝑥 𝑤)) ↔ 𝑥 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wral 3105  {crab 3109   class class class wbr 4966  cfv 6230  Basecbs 16317  lecple 16406  Posetcpo 17384  lubclub 17386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-nul 5106
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-br 4967  df-iota 6194  df-fv 6238  df-proset 17372  df-poset 17390
This theorem is referenced by:  lublecl  17433  lubid  17434
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