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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 3085 | . . . 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 29870 | . . . . . . 7 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑆‘𝑥) ·ih 𝑥) ∈ ℝ) | |
3 | hmopre 29870 | . . . . . . 7 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑥) ∈ ℝ) | |
4 | hmopre 29870 | . . . . . . 7 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈‘𝑥) ·ih 𝑥) ∈ ℝ) | |
5 | letr 10824 | . . . . . . 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 1161 | . . . . . 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 1473 | . . . . 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 3092 | . . . 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 246 | . . 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 30071 | . . . . 5 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → (𝑆 ≤op 𝑇 ↔ ∀𝑥 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑇‘𝑥) ·ih 𝑥))) | |
11 | 10 | 3adant3 1133 | . . . 4 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑆 ≤op 𝑇 ↔ ∀𝑥 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑇‘𝑥) ·ih 𝑥))) |
12 | leop2 30071 | . . . . 5 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) | |
13 | 12 | 3adant1 1131 | . . . 4 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) |
14 | 11, 13 | anbi12d 634 | . . 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 30071 | . . . 4 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑆 ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑥) ≤ ((𝑈‘𝑥) ·ih 𝑥))) | |
16 | 15 | 3adant2 1132 | . . 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 297 | . 2 ⊢ ((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((𝑆 ≤op 𝑇 ∧ 𝑇 ≤op 𝑈) → 𝑆 ≤op 𝑈)) |
18 | 17 | imp 410 | 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 209 ∧ wa 399 ∧ w3a 1088 ∈ wcel 2114 ∀wral 3054 class class class wbr 5040 ‘cfv 6349 (class class class)co 7182 ℝcr 10626 ≤ cle 10766 ℋchba 28866 ·ih csp 28869 HrmOpcho 28897 ≤op cleo 28905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-inf2 9189 ax-cc 9947 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 ax-addf 10706 ax-mulf 10707 ax-hilex 28946 ax-hfvadd 28947 ax-hvcom 28948 ax-hvass 28949 ax-hv0cl 28950 ax-hvaddid 28951 ax-hfvmul 28952 ax-hvmulid 28953 ax-hvmulass 28954 ax-hvdistr1 28955 ax-hvdistr2 28956 ax-hvmul0 28957 ax-hfi 29026 ax-his1 29029 ax-his2 29030 ax-his3 29031 ax-his4 29032 ax-hcompl 29149 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-of 7437 df-om 7612 df-1st 7726 df-2nd 7727 df-supp 7869 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-2o 8144 df-oadd 8147 df-omul 8148 df-er 8332 df-map 8451 df-pm 8452 df-ixp 8520 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-fsupp 8919 df-fi 8960 df-sup 8991 df-inf 8992 df-oi 9059 df-card 9453 df-acn 9456 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-q 12443 df-rp 12485 df-xneg 12602 df-xadd 12603 df-xmul 12604 df-ioo 12837 df-ico 12839 df-icc 12840 df-fz 12994 df-fzo 13137 df-fl 13265 df-seq 13473 df-exp 13534 df-hash 13795 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-clim 14947 df-rlim 14948 df-sum 15148 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-starv 16695 df-sca 16696 df-vsca 16697 df-ip 16698 df-tset 16699 df-ple 16700 df-ds 16702 df-unif 16703 df-hom 16704 df-cco 16705 df-rest 16811 df-topn 16812 df-0g 16830 df-gsum 16831 df-topgen 16832 df-pt 16833 df-prds 16836 df-xrs 16890 df-qtop 16895 df-imas 16896 df-xps 16898 df-mre 16972 df-mrc 16973 df-acs 16975 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-submnd 18085 df-mulg 18355 df-cntz 18577 df-cmn 19038 df-psmet 20221 df-xmet 20222 df-met 20223 df-bl 20224 df-mopn 20225 df-fbas 20226 df-fg 20227 df-cnfld 20230 df-top 21657 df-topon 21674 df-topsp 21696 df-bases 21709 df-cld 21782 df-ntr 21783 df-cls 21784 df-nei 21861 df-cn 21990 df-cnp 21991 df-lm 21992 df-haus 22078 df-tx 22325 df-hmeo 22518 df-fil 22609 df-fm 22701 df-flim 22702 df-flf 22703 df-xms 23085 df-ms 23086 df-tms 23087 df-cfil 24019 df-cau 24020 df-cmet 24021 df-grpo 28440 df-gid 28441 df-ginv 28442 df-gdiv 28443 df-ablo 28492 df-vc 28506 df-nv 28539 df-va 28542 df-ba 28543 df-sm 28544 df-0v 28545 df-vs 28546 df-nmcv 28547 df-ims 28548 df-dip 28648 df-ssp 28669 df-ph 28760 df-cbn 28810 df-hnorm 28915 df-hba 28916 df-hvsub 28918 df-hlim 28919 df-hcau 28920 df-sh 29154 df-ch 29168 df-oc 29199 df-ch0 29200 df-shs 29255 df-pjh 29342 df-hosum 29677 df-homul 29678 df-hodif 29679 df-h0op 29695 df-hmop 29791 df-leop 29799 |
This theorem is referenced by: (None) |
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