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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 15975 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → ≤ = ( ∘𝑟 𝑂 ∩ (𝐵 × 𝐵))) |
8 | 1, 2, 7 | syl2anc 691 | . . 3 ⊢ (𝜑 → ≤ = ( ∘𝑟 𝑂 ∩ (𝐵 × 𝐵))) |
9 | 8 | breqd 4594 | . 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 5104 | . . 3 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∘𝑟 𝑂𝐺 ↔ 𝐹( ∘𝑟 𝑂 ∩ (𝐵 × 𝐵))𝐺)) | |
13 | 10, 11, 12 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑂𝐺 ↔ 𝐹( ∘𝑟 𝑂 ∩ (𝐵 × 𝐵))𝐺)) |
14 | eqid 2610 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
15 | 3, 14, 4, 1, 2, 10 | pwselbas 15972 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝑅)) |
16 | ffn 5958 | . . . 4 ⊢ (𝐹:𝐼⟶(Base‘𝑅) → 𝐹 Fn 𝐼) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐼) |
18 | 3, 14, 4, 1, 2, 11 | pwselbas 15972 | . . . 4 ⊢ (𝜑 → 𝐺:𝐼⟶(Base‘𝑅)) |
19 | ffn 5958 | . . . 4 ⊢ (𝐺:𝐼⟶(Base‘𝑅) → 𝐺 Fn 𝐼) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
21 | inidm 3784 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
22 | eqidd 2611 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
23 | eqidd 2611 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
24 | 17, 20, 2, 2, 21, 22, 23 | ofrfval 6803 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑂𝐺 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)𝑂(𝐺‘𝑥))) |
25 | 9, 13, 24 | 3bitr2d 295 | 1 ⊢ (𝜑 → (𝐹 ≤ 𝐺 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)𝑂(𝐺‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∩ cin 3539 class class class wbr 4583 × cxp 5036 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑟 cofr 6794 Basecbs 15695 lecple 15775 ↑s cpws 15930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-prds 15931 df-pws 15933 |
This theorem is referenced by: (None) |
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