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Mirrors > Home > HSE Home > Th. List > leoptr | Structured version Visualization version GIF version |
Description: The positive operator ordering relation is transitive. Exercise 1(iv) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
leoptr | ⊢ (((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑆 ≤op 𝑇 ∧ 𝑇 ≤op 𝑈)) → 𝑆 ≤op 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.26 3137 | . . . 4 ⊢ (∀𝑥 ∈ ℋ (((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥)) ↔ (∀𝑥 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) | |
2 | hmopre 29396 | . . . . . . 7 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑆‘𝑥) ·ih 𝑥) ∈ ℝ) | |
3 | hmopre 29396 | . . . . . . 7 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑥) ∈ ℝ) | |
4 | hmopre 29396 | . . . . . . 7 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈‘𝑥) ·ih 𝑥) ∈ ℝ) | |
5 | letr 10586 | . . . . . . 7 ⊢ ((((𝑆‘𝑥) ·ih 𝑥) ∈ ℝ ∧ ((𝑇‘𝑥) ·ih 𝑥) ∈ ℝ ∧ ((𝑈‘𝑥) ·ih 𝑥) ∈ ℝ) → ((((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥)) → ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) | |
6 | 2, 3, 4, 5 | syl3an 1153 | . . . . . 6 ⊢ (((𝑆 ∈ HrmOp ∧ 𝑥 ∈ ℋ) ∧ (𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) ∧ (𝑈 ∈ HrmOp ∧ 𝑥 ∈ ℋ)) → ((((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥)) → ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) |
7 | 6 | 3anandirs 1464 | . . . . 5 ⊢ (((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ 𝑥 ∈ ℋ) → ((((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥)) → ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) |
8 | 7 | ralimdva 3144 | . . . 4 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (∀𝑥 ∈ ℋ (((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥)) → ∀𝑥 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) |
9 | 1, 8 | syl5bir 244 | . . 3 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((∀𝑥 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥)) → ∀𝑥 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) |
10 | leop2 29597 | . . . . 5 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → (𝑆 ≤op 𝑇 ↔ ∀𝑥 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑇‘𝑥) ·ih 𝑥))) | |
11 | 10 | 3adant3 1125 | . . . 4 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑆 ≤op 𝑇 ↔ ∀𝑥 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑇‘𝑥) ·ih 𝑥))) |
12 | leop2 29597 | . . . . 5 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) | |
13 | 12 | 3adant1 1123 | . . . 4 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) |
14 | 11, 13 | anbi12d 630 | . . 3 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((𝑆 ≤op 𝑇 ∧ 𝑇 ≤op 𝑈) ↔ (∀𝑥 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥)))) |
15 | leop2 29597 | . . . 4 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑆 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) | |
16 | 15 | 3adant2 1124 | . . 3 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑆 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) |
17 | 9, 14, 16 | 3imtr4d 295 | . 2 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((𝑆 ≤op 𝑇 ∧ 𝑇 ≤op 𝑈) → 𝑆 ≤op 𝑈)) |
18 | 17 | imp 407 | 1 ⊢ (((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑆 ≤op 𝑇 ∧ 𝑇 ≤op 𝑈)) → 𝑆 ≤op 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 ∈ wcel 2081 ∀wral 3105 class class class wbr 4966 ‘cfv 6230 (class class class)co 7021 ℝcr 10387 ≤ cle 10527 ℋchba 28392 ·ih csp 28395 HrmOpcho 28423 ≤op cleo 28431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-inf2 8955 ax-cc 9708 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-pre-sup 10466 ax-addf 10467 ax-mulf 10468 ax-hilex 28472 ax-hfvadd 28473 ax-hvcom 28474 ax-hvass 28475 ax-hv0cl 28476 ax-hvaddid 28477 ax-hfvmul 28478 ax-hvmulid 28479 ax-hvmulass 28480 ax-hvdistr1 28481 ax-hvdistr2 28482 ax-hvmul0 28483 ax-hfi 28552 ax-his1 28555 ax-his2 28556 ax-his3 28557 ax-his4 28558 ax-hcompl 28675 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-iin 4832 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-se 5408 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-isom 6239 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-of 7272 df-om 7442 df-1st 7550 df-2nd 7551 df-supp 7687 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-2o 7959 df-oadd 7962 df-omul 7963 df-er 8144 df-map 8263 df-pm 8264 df-ixp 8316 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-fsupp 8685 df-fi 8726 df-sup 8757 df-inf 8758 df-oi 8825 df-card 9219 df-acn 9222 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-div 11151 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-q 12203 df-rp 12245 df-xneg 12362 df-xadd 12363 df-xmul 12364 df-ioo 12597 df-ico 12599 df-icc 12600 df-fz 12748 df-fzo 12889 df-fl 13017 df-seq 13225 df-exp 13285 df-hash 13546 df-cj 14297 df-re 14298 df-im 14299 df-sqrt 14433 df-abs 14434 df-clim 14684 df-rlim 14685 df-sum 14882 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-starv 16414 df-sca 16415 df-vsca 16416 df-ip 16417 df-tset 16418 df-ple 16419 df-ds 16421 df-unif 16422 df-hom 16423 df-cco 16424 df-rest 16530 df-topn 16531 df-0g 16549 df-gsum 16550 df-topgen 16551 df-pt 16552 df-prds 16555 df-xrs 16609 df-qtop 16614 df-imas 16615 df-xps 16617 df-mre 16691 df-mrc 16692 df-acs 16694 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-submnd 17780 df-mulg 17987 df-cntz 18193 df-cmn 18640 df-psmet 20224 df-xmet 20225 df-met 20226 df-bl 20227 df-mopn 20228 df-fbas 20229 df-fg 20230 df-cnfld 20233 df-top 21191 df-topon 21208 df-topsp 21230 df-bases 21243 df-cld 21316 df-ntr 21317 df-cls 21318 df-nei 21395 df-cn 21524 df-cnp 21525 df-lm 21526 df-haus 21612 df-tx 21859 df-hmeo 22052 df-fil 22143 df-fm 22235 df-flim 22236 df-flf 22237 df-xms 22618 df-ms 22619 df-tms 22620 df-cfil 23546 df-cau 23547 df-cmet 23548 df-grpo 27966 df-gid 27967 df-ginv 27968 df-gdiv 27969 df-ablo 28018 df-vc 28032 df-nv 28065 df-va 28068 df-ba 28069 df-sm 28070 df-0v 28071 df-vs 28072 df-nmcv 28073 df-ims 28074 df-dip 28174 df-ssp 28195 df-ph 28286 df-cbn 28336 df-hnorm 28441 df-hba 28442 df-hvsub 28444 df-hlim 28445 df-hcau 28446 df-sh 28680 df-ch 28694 df-oc 28725 df-ch0 28726 df-shs 28781 df-pjh 28868 df-hosum 29203 df-homul 29204 df-hodif 29205 df-h0op 29221 df-hmop 29317 df-leop 29325 |
This theorem is referenced by: (None) |
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