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Mirrors > Home > MPE Home > Th. List > Mathboxes > ipoglblem | Structured version Visualization version GIF version |
Description: Lemma for ipoglbdm 46910 and ipoglb 46911. (Contributed by Zhi Wang, 29-Sep-2024.) |
Ref | Expression |
---|---|
ipolub.i | ⊢ 𝐼 = (toInc‘𝐹) |
ipolub.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
ipolub.s | ⊢ (𝜑 → 𝑆 ⊆ 𝐹) |
ipoglblem.l | ⊢ ≤ = (le‘𝐼) |
Ref | Expression |
---|---|
ipoglblem | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((𝑋 ⊆ ∩ 𝑆 ∧ ∀𝑧 ∈ 𝐹 (𝑧 ⊆ ∩ 𝑆 → 𝑧 ⊆ 𝑋)) ↔ (∀𝑦 ∈ 𝑆 𝑋 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssint 4924 | . . 3 ⊢ (𝑋 ⊆ ∩ 𝑆 ↔ ∀𝑦 ∈ 𝑆 𝑋 ⊆ 𝑦) | |
2 | ipolub.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
3 | 2 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑦 ∈ 𝑆) → 𝐹 ∈ 𝑉) |
4 | simplr 768 | . . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑦 ∈ 𝑆) → 𝑋 ∈ 𝐹) | |
5 | ipolub.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ 𝐹) | |
6 | 5 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑦 ∈ 𝑆) → 𝑆 ⊆ 𝐹) |
7 | simpr 486 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆) | |
8 | 6, 7 | sseldd 3944 | . . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝐹) |
9 | ipolub.i | . . . . . 6 ⊢ 𝐼 = (toInc‘𝐹) | |
10 | ipoglblem.l | . . . . . 6 ⊢ ≤ = (le‘𝐼) | |
11 | 9, 10 | ipole 18383 | . . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑋 ≤ 𝑦 ↔ 𝑋 ⊆ 𝑦)) |
12 | 3, 4, 8, 11 | syl3anc 1372 | . . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑦 ∈ 𝑆) → (𝑋 ≤ 𝑦 ↔ 𝑋 ⊆ 𝑦)) |
13 | 12 | ralbidva 3171 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → (∀𝑦 ∈ 𝑆 𝑋 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑋 ⊆ 𝑦)) |
14 | 1, 13 | bitr4id 290 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → (𝑋 ⊆ ∩ 𝑆 ↔ ∀𝑦 ∈ 𝑆 𝑋 ≤ 𝑦)) |
15 | ssint 4924 | . . . . 5 ⊢ (𝑧 ⊆ ∩ 𝑆 ↔ ∀𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦) | |
16 | 3 | adantlr 714 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑧 ∈ 𝐹) ∧ 𝑦 ∈ 𝑆) → 𝐹 ∈ 𝑉) |
17 | simplr 768 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑧 ∈ 𝐹) ∧ 𝑦 ∈ 𝑆) → 𝑧 ∈ 𝐹) | |
18 | 8 | adantlr 714 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑧 ∈ 𝐹) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝐹) |
19 | 9, 10 | ipole 18383 | . . . . . . 7 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑧 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑧 ≤ 𝑦 ↔ 𝑧 ⊆ 𝑦)) |
20 | 16, 17, 18, 19 | syl3anc 1372 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑧 ∈ 𝐹) ∧ 𝑦 ∈ 𝑆) → (𝑧 ≤ 𝑦 ↔ 𝑧 ⊆ 𝑦)) |
21 | 20 | ralbidva 3171 | . . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑧 ∈ 𝐹) → (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑧 ⊆ 𝑦)) |
22 | 15, 21 | bitr4id 290 | . . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑧 ∈ 𝐹) → (𝑧 ⊆ ∩ 𝑆 ↔ ∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦)) |
23 | 2 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑧 ∈ 𝐹) → 𝐹 ∈ 𝑉) |
24 | simpr 486 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑧 ∈ 𝐹) → 𝑧 ∈ 𝐹) | |
25 | simplr 768 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑧 ∈ 𝐹) → 𝑋 ∈ 𝐹) | |
26 | 9, 10 | ipole 18383 | . . . . . 6 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑧 ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝑧 ≤ 𝑋 ↔ 𝑧 ⊆ 𝑋)) |
27 | 23, 24, 25, 26 | syl3anc 1372 | . . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑧 ∈ 𝐹) → (𝑧 ≤ 𝑋 ↔ 𝑧 ⊆ 𝑋)) |
28 | 27 | bicomd 222 | . . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑧 ∈ 𝐹) → (𝑧 ⊆ 𝑋 ↔ 𝑧 ≤ 𝑋)) |
29 | 22, 28 | imbi12d 345 | . . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐹) ∧ 𝑧 ∈ 𝐹) → ((𝑧 ⊆ ∩ 𝑆 → 𝑧 ⊆ 𝑋) ↔ (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑋))) |
30 | 29 | ralbidva 3171 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → (∀𝑧 ∈ 𝐹 (𝑧 ⊆ ∩ 𝑆 → 𝑧 ⊆ 𝑋) ↔ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑋))) |
31 | 14, 30 | anbi12d 632 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((𝑋 ⊆ ∩ 𝑆 ∧ ∀𝑧 ∈ 𝐹 (𝑧 ⊆ ∩ 𝑆 → 𝑧 ⊆ 𝑋)) ↔ (∀𝑦 ∈ 𝑆 𝑋 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3063 ⊆ wss 3909 ∩ cint 4906 class class class wbr 5104 ‘cfv 6494 lecple 17100 toInccipo 18376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-z 12459 df-dec 12578 df-uz 12723 df-fz 13380 df-struct 16979 df-slot 17014 df-ndx 17026 df-base 17044 df-tset 17112 df-ple 17113 df-ocomp 17114 df-ipo 18377 |
This theorem is referenced by: ipoglbdm 46910 ipoglb 46911 |
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