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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 2818 | . . 3 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) | |
7 | eqid 2818 | . . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
8 | eqid 2818 | . . 3 ⊢ ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) = ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsval 16831 | . 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 16832 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (Base‘((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}))) |
11 | 6 | xpsff1o2 16830 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) |
12 | f1ocnv 6620 | . . . 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 6615 | . . 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 7180 | . 2 ⊢ (𝜑 → ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) ∈ V) | |
17 | xpsle.p | . 2 ⊢ ≤ = (le‘𝑇) | |
18 | 9, 10, 15, 16, 17 | imasless 16801 | 1 ⊢ (𝜑 → ≤ ⊆ ((𝑋 × 𝑌) × (𝑋 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 ∅c0 4288 {cpr 4559 〈cop 4563 × cxp 5546 ◡ccnv 5547 ran crn 5549 –onto→wfo 6346 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 1oc1o 8084 Basecbs 16471 Scalarcsca 16556 lecple 16560 Xscprds 16707 ×s cxps 16767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-prds 16709 df-imas 16769 df-xps 16771 |
This theorem is referenced by: (None) |
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