Theorem tosglblem 31904

Description: Lemma for tosglb 31905 and xrsclat 31941. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)

Hypotheses

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

Assertion

Ref	Expression
tosglblem	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑎 ≤ 𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 → 𝑐 ≤ 𝑎)) ↔ (∀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 tosglblem

Step	Hyp	Ref	Expression
1		tosglb.1	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Toset)
2	1	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → 𝐾 ∈ Toset)
3		tosglb.2	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
4	3	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐴 ⊆ 𝐵)
5	4	sselda 3947	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐵)
6		simplr 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → 𝑎 ∈ 𝐵)
7		tosglb.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
8		tosglb.e	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
9		tosglb.l	. . . . . . 7 ⊢ < = (lt‘𝐾)
10	7, 8, 9	tltnle 18325	. . . . . 6 ⊢ ((𝐾 ∈ Toset ∧ 𝑏 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵) → (𝑏 < 𝑎 ↔ ¬ 𝑎 ≤ 𝑏))
11	2, 5, 6, 10	syl3anc 1371	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → (𝑏 < 𝑎 ↔ ¬ 𝑎 ≤ 𝑏))
12	11	con2bid 354	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → (𝑎 ≤ 𝑏 ↔ ¬ 𝑏 < 𝑎))
13	12	ralbidva 3168	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑏 ∈ 𝐴 𝑎 ≤ 𝑏 ↔ ∀𝑏 ∈ 𝐴 ¬ 𝑏 < 𝑎))
14	3	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → 𝐴 ⊆ 𝐵)
15		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐴)
16	14, 15	sseldd 3948	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐵)
17	7, 8, 9	tltnle 18325	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Toset ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) → (𝑏 < 𝑐 ↔ ¬ 𝑐 ≤ 𝑏))
18	1, 17	syl3an1 1163	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) → (𝑏 < 𝑐 ↔ ¬ 𝑐 ≤ 𝑏))
19	18	3com23 1126	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑏 < 𝑐 ↔ ¬ 𝑐 ≤ 𝑏))
20	19	3expa 1118	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑏 < 𝑐 ↔ ¬ 𝑐 ≤ 𝑏))
21	20	con2bid 354	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑐 ≤ 𝑏 ↔ ¬ 𝑏 < 𝑐))
22	16, 21	syldan 591	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐵) ∧ 𝑏 ∈ 𝐴) → (𝑐 ≤ 𝑏 ↔ ¬ 𝑏 < 𝑐))
23	22	ralbidva 3168	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵) → (∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 ↔ ∀𝑏 ∈ 𝐴 ¬ 𝑏 < 𝑐))
24		breq1 5113	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → (𝑏 < 𝑐 ↔ 𝑑 < 𝑐))
25	24	notbid 317	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑑 → (¬ 𝑏 < 𝑐 ↔ ¬ 𝑑 < 𝑐))
26	25	cbvralvw 3223	. . . . . . . . . 10 ⊢ (∀𝑏 ∈ 𝐴 ¬ 𝑏 < 𝑐 ↔ ∀𝑑 ∈ 𝐴 ¬ 𝑑 < 𝑐)
27		ralnex 3071	. . . . . . . . . 10 ⊢ (∀𝑑 ∈ 𝐴 ¬ 𝑑 < 𝑐 ↔ ¬ ∃𝑑 ∈ 𝐴 𝑑 < 𝑐)
28	26, 27	bitri 274	. . . . . . . . 9 ⊢ (∀𝑏 ∈ 𝐴 ¬ 𝑏 < 𝑐 ↔ ¬ ∃𝑑 ∈ 𝐴 𝑑 < 𝑐)
29	23, 28	bitrdi 286	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐵) → (∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 ↔ ¬ ∃𝑑 ∈ 𝐴 𝑑 < 𝑐))
30	29	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → (∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 ↔ ¬ ∃𝑑 ∈ 𝐴 𝑑 < 𝑐))
31	1	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → 𝐾 ∈ Toset)
32		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → 𝑎 ∈ 𝐵)
33		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ 𝐵)
34	7, 8, 9	tltnle 18325	. . . . . . . . 9 ⊢ ((𝐾 ∈ Toset ∧ 𝑎 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) → (𝑎 < 𝑐 ↔ ¬ 𝑐 ≤ 𝑎))
35	31, 32, 33, 34	syl3anc 1371	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → (𝑎 < 𝑐 ↔ ¬ 𝑐 ≤ 𝑎))
36	35	con2bid 354	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → (𝑐 ≤ 𝑎 ↔ ¬ 𝑎 < 𝑐))
37	30, 36	imbi12d 344	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 → 𝑐 ≤ 𝑎) ↔ (¬ ∃𝑑 ∈ 𝐴 𝑑 < 𝑐 → ¬ 𝑎 < 𝑐)))
38		con34b 315	. . . . . 6 ⊢ ((𝑎 < 𝑐 → ∃𝑑 ∈ 𝐴 𝑑 < 𝑐) ↔ (¬ ∃𝑑 ∈ 𝐴 𝑑 < 𝑐 → ¬ 𝑎 < 𝑐))
39	37, 38	bitr4di 288	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑐 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 → 𝑐 ≤ 𝑎) ↔ (𝑎 < 𝑐 → ∃𝑑 ∈ 𝐴 𝑑 < 𝑐)))
40	39	ralbidva 3168	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 → 𝑐 ≤ 𝑎) ↔ ∀𝑐 ∈ 𝐵 (𝑎 < 𝑐 → ∃𝑑 ∈ 𝐴 𝑑 < 𝑐)))
41		breq2 5114	. . . . . 6 ⊢ (𝑏 = 𝑐 → (𝑎 < 𝑏 ↔ 𝑎 < 𝑐))
42		breq2 5114	. . . . . . 7 ⊢ (𝑏 = 𝑐 → (𝑑 < 𝑏 ↔ 𝑑 < 𝑐))
43	42	rexbidv 3171	. . . . . 6 ⊢ (𝑏 = 𝑐 → (∃𝑑 ∈ 𝐴 𝑑 < 𝑏 ↔ ∃𝑑 ∈ 𝐴 𝑑 < 𝑐))
44	41, 43	imbi12d 344	. . . . 5 ⊢ (𝑏 = 𝑐 → ((𝑎 < 𝑏 → ∃𝑑 ∈ 𝐴 𝑑 < 𝑏) ↔ (𝑎 < 𝑐 → ∃𝑑 ∈ 𝐴 𝑑 < 𝑐)))
45	44	cbvralvw 3223	. . . 4 ⊢ (∀𝑏 ∈ 𝐵 (𝑎 < 𝑏 → ∃𝑑 ∈ 𝐴 𝑑 < 𝑏) ↔ ∀𝑐 ∈ 𝐵 (𝑎 < 𝑐 → ∃𝑑 ∈ 𝐴 𝑑 < 𝑐))
46	40, 45	bitr4di 288	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 → 𝑐 ≤ 𝑎) ↔ ∀𝑏 ∈ 𝐵 (𝑎 < 𝑏 → ∃𝑑 ∈ 𝐴 𝑑 < 𝑏)))
47	13, 46	anbi12d 631	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑎 ≤ 𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 → 𝑐 ≤ 𝑎)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ 𝐵 (𝑎 < 𝑏 → ∃𝑑 ∈ 𝐴 𝑑 < 𝑏))))
48		vex 3450	. . . . . 6 ⊢ 𝑎 ∈ V
49		vex 3450	. . . . . 6 ⊢ 𝑏 ∈ V
50	48, 49	brcnv 5843	. . . . 5 ⊢ (𝑎◡ < 𝑏 ↔ 𝑏 < 𝑎)
51	50	notbii 319	. . . 4 ⊢ (¬ 𝑎◡ < 𝑏 ↔ ¬ 𝑏 < 𝑎)
52	51	ralbii 3092	. . 3 ⊢ (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ↔ ∀𝑏 ∈ 𝐴 ¬ 𝑏 < 𝑎)
53	49, 48	brcnv 5843	. . . . 5 ⊢ (𝑏◡ < 𝑎 ↔ 𝑎 < 𝑏)
54		vex 3450	. . . . . . 7 ⊢ 𝑑 ∈ V
55	49, 54	brcnv 5843	. . . . . 6 ⊢ (𝑏◡ < 𝑑 ↔ 𝑑 < 𝑏)
56	55	rexbii 3093	. . . . 5 ⊢ (∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑 ↔ ∃𝑑 ∈ 𝐴 𝑑 < 𝑏)
57	53, 56	imbi12i 350	. . . 4 ⊢ ((𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑) ↔ (𝑎 < 𝑏 → ∃𝑑 ∈ 𝐴 𝑑 < 𝑏))
58	57	ralbii 3092	. . 3 ⊢ (∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑) ↔ ∀𝑏 ∈ 𝐵 (𝑎 < 𝑏 → ∃𝑑 ∈ 𝐴 𝑑 < 𝑏))
59	52, 58	anbi12i 627	. 2 ⊢ ((∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ 𝐵 (𝑎 < 𝑏 → ∃𝑑 ∈ 𝐴 𝑑 < 𝑏)))
60	47, 59	bitr4di 288	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑎 ≤ 𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 → 𝑐 ≤ 𝑎)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069   ⊆ wss 3913   class class class wbr 5110  ^◡ccnv 5637  ‘cfv 6501  Basecbs 17094  lecple 17154  ltcplt 18211  Tosetctos 18319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-proset 18198  df-poset 18216  df-plt 18233  df-toset 18320

This theorem is referenced by:  tosglb  31905  xrsclat  31941