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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 5150 | . . 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 7229 | . . . . . . 7 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V) | |
7 | 4, 5, 6 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V) |
8 | prdsbasmpt.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
9 | 4 | fndmd 6660 | . . . . . 6 ⊢ (𝜑 → dom 𝑅 = 𝐼) |
10 | prdsleval.l | . . . . . 6 ⊢ ≤ = (le‘𝑌) | |
11 | 2, 3, 7, 8, 9, 10 | prdsle 17463 | . . . . 5 ⊢ (𝜑 → ≤ = {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) |
12 | vex 3465 | . . . . . . . 8 ⊢ 𝑓 ∈ V | |
13 | vex 3465 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
14 | 12, 13 | prss 4825 | . . . . . . 7 ⊢ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ↔ {𝑓, 𝑔} ⊆ 𝐵) |
15 | 14 | anbi1i 622 | . . . . . 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))) |
16 | 15 | opabbii 5216 | . . . . 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))} |
17 | 11, 16 | eqtr4di 2783 | . . . 4 ⊢ (𝜑 → ≤ = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) |
18 | 17 | eleq2d 2811 | . . 3 ⊢ (𝜑 → (〈𝐹, 𝐺〉 ∈ ≤ ↔ 〈𝐹, 𝐺〉 ∈ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})) |
19 | 1, 18 | bitrid 282 | . 2 ⊢ (𝜑 → (𝐹 ≤ 𝐺 ↔ 〈𝐹, 𝐺〉 ∈ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})) |
20 | prdsplusgval.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
21 | prdsplusgval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
22 | fveq1 6895 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
23 | fveq1 6895 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥)) | |
24 | 22, 23 | breqan12d 5165 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥) ↔ (𝐹‘𝑥)(le‘(𝑅‘𝑥))(𝐺‘𝑥))) |
25 | 24 | ralbidv 3167 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥) ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)(le‘(𝑅‘𝑥))(𝐺‘𝑥))) |
26 | 25 | opelopab2a 5537 | . . 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))) |
27 | 20, 21, 26 | syl2anc 582 | . 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))) |
28 | 19, 27 | bitrd 278 | 1 ⊢ (𝜑 → (𝐹 ≤ 𝐺 ↔ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥)(le‘(𝑅‘𝑥))(𝐺‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 Vcvv 3461 ⊆ wss 3944 {cpr 4632 〈cop 4636 class class class wbr 5149 {copab 5211 Fn wfn 6544 ‘cfv 6549 (class class class)co 7419 Basecbs 17199 lecple 17259 Xscprds 17446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9472 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17135 df-slot 17170 df-ndx 17182 df-base 17200 df-plusg 17265 df-mulr 17266 df-sca 17268 df-vsca 17269 df-ip 17270 df-tset 17271 df-ple 17272 df-ds 17274 df-hom 17276 df-cco 17277 df-prds 17448 |
This theorem is referenced by: xpsle 17580 |
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