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Mirrors > Home > MPE Home > Th. List > prdsleval | Structured version Visualization version GIF version |
Description: Value of the product ordering in a structure product. (Contributed by Mario Carneiro, 15-Aug-2015.) |
Ref | Expression |
---|---|
prdsbasmpt.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdsbasmpt.b | ⊢ 𝐵 = (Base‘𝑌) |
prdsbasmpt.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsbasmpt.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdsbasmpt.r | ⊢ (𝜑 → 𝑅 Fn 𝐼) |
prdsplusgval.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
prdsplusgval.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
prdsleval.l | ⊢ ≤ = (le‘𝑌) |
Ref | Expression |
---|---|
prdsleval | ⊢ (𝜑 → (𝐹 ≤ 𝐺 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)(le‘(𝑅‘𝑥))(𝐺‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 4654 | . . 3 ⊢ (𝐹 ≤ 𝐺 ↔ 〈𝐹, 𝐺〉 ∈ ≤ ) | |
2 | prdsbasmpt.y | . . . . . 6 ⊢ 𝑌 = (𝑆Xs𝑅) | |
3 | prdsbasmpt.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | prdsbasmpt.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 Fn 𝐼) | |
5 | prdsbasmpt.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
6 | fnex 6481 | . . . . . . 7 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V) | |
7 | 4, 5, 6 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V) |
8 | prdsbasmpt.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
9 | fndm 5990 | . . . . . . 7 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼) | |
10 | 4, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝑅 = 𝐼) |
11 | prdsleval.l | . . . . . 6 ⊢ ≤ = (le‘𝑌) | |
12 | 2, 3, 7, 8, 10, 11 | prdsle 16122 | . . . . 5 ⊢ (𝜑 → ≤ = {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) |
13 | vex 3203 | . . . . . . . 8 ⊢ 𝑓 ∈ V | |
14 | vex 3203 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
15 | 13, 14 | prss 4351 | . . . . . . 7 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ↔ {𝑓, 𝑔} ⊆ 𝐵) |
16 | 15 | anbi1i 731 | . . . . . 6 ⊢ (((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥)) ↔ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))) |
17 | 16 | opabbii 4717 | . . . . 5 ⊢ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} = {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} |
18 | 12, 17 | syl6eqr 2674 | . . . 4 ⊢ (𝜑 → ≤ = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) |
19 | 18 | eleq2d 2687 | . . 3 ⊢ (𝜑 → (〈𝐹, 𝐺〉 ∈ ≤ ↔ 〈𝐹, 𝐺〉 ∈ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})) |
20 | 1, 19 | syl5bb 272 | . 2 ⊢ (𝜑 → (𝐹 ≤ 𝐺 ↔ 〈𝐹, 𝐺〉 ∈ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})) |
21 | prdsplusgval.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
22 | prdsplusgval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
23 | fveq1 6190 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
24 | fveq1 6190 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥)) | |
25 | 23, 24 | breqan12d 4669 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥) ↔ (𝐹‘𝑥)(le‘(𝑅‘𝑥))(𝐺‘𝑥))) |
26 | 25 | ralbidv 2986 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥) ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)(le‘(𝑅‘𝑥))(𝐺‘𝑥))) |
27 | 26 | opelopab2a 4990 | . . 3 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (〈𝐹, 𝐺〉 ∈ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)(le‘(𝑅‘𝑥))(𝐺‘𝑥))) |
28 | 21, 22, 27 | syl2anc 693 | . 2 ⊢ (𝜑 → (〈𝐹, 𝐺〉 ∈ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)(le‘(𝑅‘𝑥))(𝐺‘𝑥))) |
29 | 20, 28 | bitrd 268 | 1 ⊢ (𝜑 → (𝐹 ≤ 𝐺 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)(le‘(𝑅‘𝑥))(𝐺‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 {cpr 4179 〈cop 4183 class class class wbr 4653 {copab 4712 dom cdm 5114 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 Xscprds 16106 |
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-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-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 |
This theorem is referenced by: xpsle 16241 |
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