Theorem toslublem 28998

Description: Lemma for toslub 28999 and xrsclat 29011. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)

Hypotheses

Ref	Expression
toslub.b	⊢ 𝐵 = (Base‘𝐾)
toslub.l	⊢ < = (lt‘𝐾)
toslub.1	⊢ (𝜑 → 𝐾 ∈ Toset)
toslub.2	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
toslub.e	⊢ ≤ = (le‘𝐾)

Assertion

Ref	Expression
toslublem	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑐 → 𝑎 ≤ 𝑐)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏 < 𝑑))))

Distinct variable groups:   𝑎,𝑏,𝑐,𝑑, <   𝐴,𝑎,𝑏,𝑐,𝑑   𝐵,𝑎,𝑏,𝑐,𝑑   𝐾,𝑎,𝑏,𝑐   𝜑,𝑎,𝑏,𝑐

Allowed substitution hints:   𝜑(𝑑)   𝐾(𝑑)   ≤ (𝑎,𝑏,𝑐,𝑑)

Proof of Theorem toslublem

Step	Hyp	Ref	Expression
1		toslub.1	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Toset)
2	1	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → 𝐾 ∈ Toset)
3		simplr 788	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → 𝑎 ∈ 𝐵)
4		toslub.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
6	5	sselda 3568	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐵)
7		toslub.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
8		toslub.e	. . . . . 6 ⊢ ≤ = (le‘𝐾)
9		toslub.l	. . . . . 6 ⊢ < = (lt‘𝐾)
10	7, 8, 9	tltnle 28993	. . . . 5 ⊢ ((𝐾 ∈ Toset ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑎))
11	2, 3, 6, 10	syl3anc 1318	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → (𝑎 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑎))
12	11	con2bid 343	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → (𝑏 ≤ 𝑎 ↔ ¬ 𝑎 < 𝑏))
13	12	ralbidva 2968	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏))
14	4	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → 𝐴 ⊆ 𝐵)
15		simpr 476	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐴)
16	14, 15	sseldd 3569	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐵)
17	7, 8, 9	tltnle 28993	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Toset ∧ 𝑐 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑐 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑐))
18	1, 17	syl3an1 1351	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑐 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑐))
19	18	3expa 1257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑐 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑐))
20	19	con2bid 343	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑏 ≤ 𝑐 ↔ ¬ 𝑐 < 𝑏))
21	16, 20	syldan 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → (𝑏 ≤ 𝑐 ↔ ¬ 𝑐 < 𝑏))
22	21	ralbidva 2968	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵) → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑐 ↔ ∀𝑏 ∈ 𝐴 ¬ 𝑐 < 𝑏))
23		breq2 4587	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑑 → (𝑐 < 𝑏 ↔ 𝑐 < 𝑑))
24	23	notbid 307	. . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → (¬ 𝑐 < 𝑏 ↔ ¬ 𝑐 < 𝑑))
25	24	cbvralv 3147	. . . . . . . . 9 ⊢ (∀𝑏 ∈ 𝐴 ¬ 𝑐 < 𝑏 ↔ ∀𝑑 ∈ 𝐴 ¬ 𝑐 < 𝑑)
26		ralnex 2975	. . . . . . . . 9 ⊢ (∀𝑑 ∈ 𝐴 ¬ 𝑐 < 𝑑 ↔ ¬ ∃𝑑 ∈ 𝐴 𝑐 < 𝑑)
27	25, 26	bitri 263	. . . . . . . 8 ⊢ (∀𝑏 ∈ 𝐴 ¬ 𝑐 < 𝑏 ↔ ¬ ∃𝑑 ∈ 𝐴 𝑐 < 𝑑)
28	22, 27	syl6bb 275	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵) → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑐 ↔ ¬ ∃𝑑 ∈ 𝐴 𝑐 < 𝑑))
29	28	adantlr 747	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑐 ↔ ¬ ∃𝑑 ∈ 𝐴 𝑐 < 𝑑))
30	1	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → 𝐾 ∈ Toset)
31		simpr 476	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ 𝐵)
32		simplr 788	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → 𝑎 ∈ 𝐵)
33	7, 8, 9	tltnle 28993	. . . . . . . 8 ⊢ ((𝐾 ∈ Toset ∧ 𝑐 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑐 < 𝑎 ↔ ¬ 𝑎 ≤ 𝑐))
34	30, 31, 32, 33	syl3anc 1318	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → (𝑐 < 𝑎 ↔ ¬ 𝑎 ≤ 𝑐))
35	34	con2bid 343	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → (𝑎 ≤ 𝑐 ↔ ¬ 𝑐 < 𝑎))
36	29, 35	imbi12d 333	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑐 → 𝑎 ≤ 𝑐) ↔ (¬ ∃𝑑 ∈ 𝐴 𝑐 < 𝑑 → ¬ 𝑐 < 𝑎)))
37		con34b 305	. . . . 5 ⊢ ((𝑐 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑐 < 𝑑) ↔ (¬ ∃𝑑 ∈ 𝐴 𝑐 < 𝑑 → ¬ 𝑐 < 𝑎))
38	36, 37	syl6bbr 277	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑐 → 𝑎 ≤ 𝑐) ↔ (𝑐 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑐 < 𝑑)))
39	38	ralbidva 2968	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑐 → 𝑎 ≤ 𝑐) ↔ ∀𝑐 ∈ 𝐵 (𝑐 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑐 < 𝑑)))
40		breq1 4586	. . . . 5 ⊢ (𝑏 = 𝑐 → (𝑏 < 𝑎 ↔ 𝑐 < 𝑎))
41		breq1 4586	. . . . . 6 ⊢ (𝑏 = 𝑐 → (𝑏 < 𝑑 ↔ 𝑐 < 𝑑))
42	41	rexbidv 3034	. . . . 5 ⊢ (𝑏 = 𝑐 → (∃𝑑 ∈ 𝐴 𝑏 < 𝑑 ↔ ∃𝑑 ∈ 𝐴 𝑐 < 𝑑))
43	40, 42	imbi12d 333	. . . 4 ⊢ (𝑏 = 𝑐 → ((𝑏 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏 < 𝑑) ↔ (𝑐 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑐 < 𝑑)))
44	43	cbvralv 3147	. . 3 ⊢ (∀𝑏 ∈ 𝐵 (𝑏 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏 < 𝑑) ↔ ∀𝑐 ∈ 𝐵 (𝑐 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑐 < 𝑑))
45	39, 44	syl6bbr 277	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑐 → 𝑎 ≤ 𝑐) ↔ ∀𝑏 ∈ 𝐵 (𝑏 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏 < 𝑑)))
46	13, 45	anbi12d 743	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑐 → 𝑎 ≤ 𝑐)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏 < 𝑑))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  ltcplt 16764  Tosetctos 16856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-preset 16751  df-poset 16769  df-plt 16781  df-toset 16857

This theorem is referenced by:  toslub  28999  xrsclat  29011