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Mirrors > Home > MPE Home > Th. List > xpsless | Structured version Visualization version GIF version |
Description: Closure of the ordering in a binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015.) |
Ref | Expression |
---|---|
xpsle.t | ⊢ 𝑇 = (𝑅 ×s 𝑆) |
xpsle.x | ⊢ 𝑋 = (Base‘𝑅) |
xpsle.y | ⊢ 𝑌 = (Base‘𝑆) |
xpsle.1 | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
xpsle.2 | ⊢ (𝜑 → 𝑆 ∈ 𝑊) |
xpsle.p | ⊢ ≤ = (le‘𝑇) |
Ref | Expression |
---|---|
xpsless | ⊢ (𝜑 → ≤ ⊆ ((𝑋 × 𝑌) × (𝑋 × 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsle.t | . . 3 ⊢ 𝑇 = (𝑅 ×s 𝑆) | |
2 | xpsle.x | . . 3 ⊢ 𝑋 = (Base‘𝑅) | |
3 | xpsle.y | . . 3 ⊢ 𝑌 = (Base‘𝑆) | |
4 | xpsle.1 | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
5 | xpsle.2 | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑊) | |
6 | eqid 2622 | . . 3 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) | |
7 | eqid 2622 | . . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
8 | eqid 2622 | . . 3 ⊢ ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) = ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsval 16232 | . 2 ⊢ (𝜑 → 𝑇 = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))) |
10 | 1, 2, 3, 4, 5, 6, 7, 8 | xpslem 16233 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) = (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))) |
11 | 6 | xpsff1o2 16231 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) |
12 | f1ocnv 6149 | . . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌)) | |
13 | 11, 12 | mp1i 13 | . . 3 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌)) |
14 | f1ofo 6144 | . . 3 ⊢ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–onto→(𝑋 × 𝑌)) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–onto→(𝑋 × 𝑌)) |
16 | ovexd 6680 | . 2 ⊢ (𝜑 → ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) ∈ V) | |
17 | xpsle.p | . 2 ⊢ ≤ = (le‘𝑇) | |
18 | 9, 10, 15, 16, 17 | imasless 16200 | 1 ⊢ (𝜑 → ≤ ⊆ ((𝑋 × 𝑌) × (𝑋 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 {csn 4177 × cxp 5112 ◡ccnv 5113 ran crn 5115 –onto→wfo 5886 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 +𝑐 ccda 8989 Basecbs 15857 Scalarcsca 15944 lecple 15948 Xscprds 16106 ×s cxps 16166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-prds 16108 df-imas 16168 df-xps 16170 |
This theorem is referenced by: (None) |
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