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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 2821 | . . 3 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) | |
7 | eqid 2821 | . . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
8 | eqid 2821 | . . 3 ⊢ ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) = ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsval 16842 | . 2 ⊢ (𝜑 → 𝑇 = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) “s ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}))) |
10 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsrnbas 16843 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (Base‘((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}))) |
11 | 6 | xpsff1o2 16841 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) |
12 | f1ocnv 6626 | . . . 4 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–1-1-onto→(𝑋 × 𝑌)) | |
13 | 11, 12 | mp1i 13 | . . 3 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–1-1-onto→(𝑋 × 𝑌)) |
14 | f1ofo 6621 | . . 3 ⊢ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–1-1-onto→(𝑋 × 𝑌) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–onto→(𝑋 × 𝑌)) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–onto→(𝑋 × 𝑌)) |
16 | ovexd 7190 | . 2 ⊢ (𝜑 → ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) ∈ V) | |
17 | xpsle.p | . 2 ⊢ ≤ = (le‘𝑇) | |
18 | 9, 10, 15, 16, 17 | imasless 16812 | 1 ⊢ (𝜑 → ≤ ⊆ ((𝑋 × 𝑌) × (𝑋 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 {cpr 4568 〈cop 4572 × cxp 5552 ◡ccnv 5553 ran crn 5555 –onto→wfo 6352 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 1oc1o 8094 Basecbs 16482 Scalarcsca 16567 lecple 16571 Xscprds 16718 ×s cxps 16778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-hom 16588 df-cco 16589 df-prds 16720 df-imas 16780 df-xps 16782 |
This theorem is referenced by: (None) |
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