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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihcnvord | Structured version Visualization version GIF version |
Description: Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.) |
Ref | Expression |
---|---|
dihcnvord.l | ⊢ ≤ = (le‘𝐾) |
dihcnvord.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihcnvord.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihcnvord.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihcnvord.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) |
dihcnvord.y | ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) |
Ref | Expression |
---|---|
dihcnvord | ⊢ (𝜑 → ((◡𝐼‘𝑋) ≤ (◡𝐼‘𝑌) ↔ 𝑋 ⊆ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihcnvord.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dihcnvord.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) | |
3 | eqid 2731 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | dihcnvord.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | dihcnvord.i | . . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | 3, 4, 5 | dihcnvcl 40605 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
7 | 1, 2, 6 | syl2anc 583 | . . 3 ⊢ (𝜑 → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
8 | dihcnvord.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) | |
9 | 3, 4, 5 | dihcnvcl 40605 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → (◡𝐼‘𝑌) ∈ (Base‘𝐾)) |
10 | 1, 8, 9 | syl2anc 583 | . . 3 ⊢ (𝜑 → (◡𝐼‘𝑌) ∈ (Base‘𝐾)) |
11 | dihcnvord.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
12 | 3, 11, 4, 5 | dihord 40598 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (◡𝐼‘𝑋) ∈ (Base‘𝐾) ∧ (◡𝐼‘𝑌) ∈ (Base‘𝐾)) → ((𝐼‘(◡𝐼‘𝑋)) ⊆ (𝐼‘(◡𝐼‘𝑌)) ↔ (◡𝐼‘𝑋) ≤ (◡𝐼‘𝑌))) |
13 | 1, 7, 10, 12 | syl3anc 1370 | . 2 ⊢ (𝜑 → ((𝐼‘(◡𝐼‘𝑋)) ⊆ (𝐼‘(◡𝐼‘𝑌)) ↔ (◡𝐼‘𝑋) ≤ (◡𝐼‘𝑌))) |
14 | 4, 5 | dihcnvid2 40607 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝐼‘(◡𝐼‘𝑋)) = 𝑋) |
15 | 1, 2, 14 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘𝑋)) = 𝑋) |
16 | 4, 5 | dihcnvid2 40607 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → (𝐼‘(◡𝐼‘𝑌)) = 𝑌) |
17 | 1, 8, 16 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘𝑌)) = 𝑌) |
18 | 15, 17 | sseq12d 4015 | . 2 ⊢ (𝜑 → ((𝐼‘(◡𝐼‘𝑋)) ⊆ (𝐼‘(◡𝐼‘𝑌)) ↔ 𝑋 ⊆ 𝑌)) |
19 | 13, 18 | bitr3d 281 | 1 ⊢ (𝜑 → ((◡𝐼‘𝑋) ≤ (◡𝐼‘𝑌) ↔ 𝑋 ⊆ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ⊆ wss 3948 class class class wbr 5148 ◡ccnv 5675 ran crn 5677 ‘cfv 6543 Basecbs 17151 lecple 17211 HLchlt 38683 LHypclh 39318 DIsoHcdih 40562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-riotaBAD 38286 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-undef 8264 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cntz 19229 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20946 df-oposet 38509 df-ol 38511 df-oml 38512 df-covers 38599 df-ats 38600 df-atl 38631 df-cvlat 38655 df-hlat 38684 df-llines 38832 df-lplanes 38833 df-lvols 38834 df-lines 38835 df-psubsp 38837 df-pmap 38838 df-padd 39130 df-lhyp 39322 df-laut 39323 df-ldil 39438 df-ltrn 39439 df-trl 39493 df-tendo 40089 df-edring 40091 df-disoa 40363 df-dvech 40413 df-dib 40473 df-dic 40507 df-dih 40563 |
This theorem is referenced by: dihoml4c 40710 djhcvat42 40749 |
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