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Mirrors > Home > MPE Home > Th. List > pwsleval | Structured version Visualization version GIF version |
Description: Ordering in a structure power. (Contributed by Mario Carneiro, 16-Aug-2015.) |
Ref | Expression |
---|---|
pwsle.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsle.v | ⊢ 𝐵 = (Base‘𝑌) |
pwsle.o | ⊢ 𝑂 = (le‘𝑅) |
pwsle.l | ⊢ ≤ = (le‘𝑌) |
pwsleval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
pwsleval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
pwsleval.a | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
pwsleval.b | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
Ref | Expression |
---|---|
pwsleval | ⊢ (𝜑 → (𝐹 ≤ 𝐺 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)𝑂(𝐺‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsleval.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
2 | pwsleval.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
3 | pwsle.y | . . . . 5 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
4 | pwsle.v | . . . . 5 ⊢ 𝐵 = (Base‘𝑌) | |
5 | pwsle.o | . . . . 5 ⊢ 𝑂 = (le‘𝑅) | |
6 | pwsle.l | . . . . 5 ⊢ ≤ = (le‘𝑌) | |
7 | 3, 4, 5, 6 | pwsle 16757 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → ≤ = ( ∘r 𝑂 ∩ (𝐵 × 𝐵))) |
8 | 1, 2, 7 | syl2anc 587 | . . 3 ⊢ (𝜑 → ≤ = ( ∘r 𝑂 ∩ (𝐵 × 𝐵))) |
9 | 8 | breqd 5041 | . 2 ⊢ (𝜑 → (𝐹 ≤ 𝐺 ↔ 𝐹( ∘r 𝑂 ∩ (𝐵 × 𝐵))𝐺)) |
10 | pwsleval.a | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
11 | pwsleval.b | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
12 | brinxp 5594 | . . 3 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∘r 𝑂𝐺 ↔ 𝐹( ∘r 𝑂 ∩ (𝐵 × 𝐵))𝐺)) | |
13 | 10, 11, 12 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐹 ∘r 𝑂𝐺 ↔ 𝐹( ∘r 𝑂 ∩ (𝐵 × 𝐵))𝐺)) |
14 | eqid 2798 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
15 | 3, 14, 4, 1, 2, 10 | pwselbas 16754 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝑅)) |
16 | 15 | ffnd 6488 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐼) |
17 | 3, 14, 4, 1, 2, 11 | pwselbas 16754 | . . . 4 ⊢ (𝜑 → 𝐺:𝐼⟶(Base‘𝑅)) |
18 | 17 | ffnd 6488 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
19 | inidm 4145 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
20 | eqidd 2799 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
21 | eqidd 2799 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
22 | 16, 18, 2, 2, 19, 20, 21 | ofrfval 7397 | . 2 ⊢ (𝜑 → (𝐹 ∘r 𝑂𝐺 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)𝑂(𝐺‘𝑥))) |
23 | 9, 13, 22 | 3bitr2d 310 | 1 ⊢ (𝜑 → (𝐹 ≤ 𝐺 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)𝑂(𝐺‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∩ cin 3880 class class class wbr 5030 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ∘r cofr 7388 Basecbs 16475 lecple 16564 ↑s cpws 16712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-prds 16713 df-pws 16715 |
This theorem is referenced by: (None) |
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