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Mirrors > Home > HSE Home > Th. List > leopadd | Structured version Visualization version GIF version |
Description: The sum of two positive operators is positive. Exercise 1(i) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
leopadd | ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ ( 0hop ≤op 𝑇 ∧ 0hop ≤op 𝑈)) → 0hop ≤op (𝑇 +op 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.26 3110 | . . . 4 ⊢ (∀𝑥 ∈ ℋ (0 ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ 0 ≤ ((𝑈‘𝑥) ·ih 𝑥)) ↔ (∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ ∀𝑥 ∈ ℋ 0 ≤ ((𝑈‘𝑥) ·ih 𝑥))) | |
2 | hmopre 31444 | . . . . . . . 8 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑥) ∈ ℝ) | |
3 | hmopre 31444 | . . . . . . . 8 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈‘𝑥) ·ih 𝑥) ∈ ℝ) | |
4 | addge0 11708 | . . . . . . . . 9 ⊢ (((((𝑇‘𝑥) ·ih 𝑥) ∈ ℝ ∧ ((𝑈‘𝑥) ·ih 𝑥) ∈ ℝ) ∧ (0 ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ 0 ≤ ((𝑈‘𝑥) ·ih 𝑥))) → 0 ≤ (((𝑇‘𝑥) ·ih 𝑥) + ((𝑈‘𝑥) ·ih 𝑥))) | |
5 | 4 | ex 412 | . . . . . . . 8 ⊢ ((((𝑇‘𝑥) ·ih 𝑥) ∈ ℝ ∧ ((𝑈‘𝑥) ·ih 𝑥) ∈ ℝ) → ((0 ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ 0 ≤ ((𝑈‘𝑥) ·ih 𝑥)) → 0 ≤ (((𝑇‘𝑥) ·ih 𝑥) + ((𝑈‘𝑥) ·ih 𝑥)))) |
6 | 2, 3, 5 | syl2an 595 | . . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) ∧ (𝑈 ∈ HrmOp ∧ 𝑥 ∈ ℋ)) → ((0 ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ 0 ≤ ((𝑈‘𝑥) ·ih 𝑥)) → 0 ≤ (((𝑇‘𝑥) ·ih 𝑥) + ((𝑈‘𝑥) ·ih 𝑥)))) |
7 | 6 | anandirs 676 | . . . . . 6 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ 𝑥 ∈ ℋ) → ((0 ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ 0 ≤ ((𝑈‘𝑥) ·ih 𝑥)) → 0 ≤ (((𝑇‘𝑥) ·ih 𝑥) + ((𝑈‘𝑥) ·ih 𝑥)))) |
8 | hmopf 31395 | . . . . . . . . 9 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ) | |
9 | hmopf 31395 | . . . . . . . . 9 ⊢ (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ) | |
10 | 8, 9 | anim12i 612 | . . . . . . . 8 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) |
11 | hosval 31261 | . . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑈)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑈‘𝑥))) | |
12 | 11 | oveq1d 7427 | . . . . . . . . . 10 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑇 +op 𝑈)‘𝑥) ·ih 𝑥) = (((𝑇‘𝑥) +ℎ (𝑈‘𝑥)) ·ih 𝑥)) |
13 | 12 | 3expa 1117 | . . . . . . . . 9 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝑇 +op 𝑈)‘𝑥) ·ih 𝑥) = (((𝑇‘𝑥) +ℎ (𝑈‘𝑥)) ·ih 𝑥)) |
14 | ffvelcdm 7083 | . . . . . . . . . . 11 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ) | |
15 | 14 | adantlr 712 | . . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ) |
16 | ffvelcdm 7083 | . . . . . . . . . . 11 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑈‘𝑥) ∈ ℋ) | |
17 | 16 | adantll 711 | . . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑈‘𝑥) ∈ ℋ) |
18 | simpr 484 | . . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → 𝑥 ∈ ℋ) | |
19 | ax-his2 30604 | . . . . . . . . . 10 ⊢ (((𝑇‘𝑥) ∈ ℋ ∧ (𝑈‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) → (((𝑇‘𝑥) +ℎ (𝑈‘𝑥)) ·ih 𝑥) = (((𝑇‘𝑥) ·ih 𝑥) + ((𝑈‘𝑥) ·ih 𝑥))) | |
20 | 15, 17, 18, 19 | syl3anc 1370 | . . . . . . . . 9 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝑇‘𝑥) +ℎ (𝑈‘𝑥)) ·ih 𝑥) = (((𝑇‘𝑥) ·ih 𝑥) + ((𝑈‘𝑥) ·ih 𝑥))) |
21 | 13, 20 | eqtrd 2771 | . . . . . . . 8 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝑇 +op 𝑈)‘𝑥) ·ih 𝑥) = (((𝑇‘𝑥) ·ih 𝑥) + ((𝑈‘𝑥) ·ih 𝑥))) |
22 | 10, 21 | sylan 579 | . . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ 𝑥 ∈ ℋ) → (((𝑇 +op 𝑈)‘𝑥) ·ih 𝑥) = (((𝑇‘𝑥) ·ih 𝑥) + ((𝑈‘𝑥) ·ih 𝑥))) |
23 | 22 | breq2d 5160 | . . . . . 6 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ 𝑥 ∈ ℋ) → (0 ≤ (((𝑇 +op 𝑈)‘𝑥) ·ih 𝑥) ↔ 0 ≤ (((𝑇‘𝑥) ·ih 𝑥) + ((𝑈‘𝑥) ·ih 𝑥)))) |
24 | 7, 23 | sylibrd 259 | . . . . 5 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ 𝑥 ∈ ℋ) → ((0 ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ 0 ≤ ((𝑈‘𝑥) ·ih 𝑥)) → 0 ≤ (((𝑇 +op 𝑈)‘𝑥) ·ih 𝑥))) |
25 | 24 | ralimdva 3166 | . . . 4 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (∀𝑥 ∈ ℋ (0 ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ 0 ≤ ((𝑈‘𝑥) ·ih 𝑥)) → ∀𝑥 ∈ ℋ 0 ≤ (((𝑇 +op 𝑈)‘𝑥) ·ih 𝑥))) |
26 | 1, 25 | biimtrrid 242 | . . 3 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ ∀𝑥 ∈ ℋ 0 ≤ ((𝑈‘𝑥) ·ih 𝑥)) → ∀𝑥 ∈ ℋ 0 ≤ (((𝑇 +op 𝑈)‘𝑥) ·ih 𝑥))) |
27 | leoppos 31647 | . . . 4 ⊢ (𝑇 ∈ HrmOp → ( 0hop ≤op 𝑇 ↔ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥))) | |
28 | leoppos 31647 | . . . 4 ⊢ (𝑈 ∈ HrmOp → ( 0hop ≤op 𝑈 ↔ ∀𝑥 ∈ ℋ 0 ≤ ((𝑈‘𝑥) ·ih 𝑥))) | |
29 | 27, 28 | bi2anan9 636 | . . 3 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (( 0hop ≤op 𝑇 ∧ 0hop ≤op 𝑈) ↔ (∀𝑥 ∈ ℋ 0 ≤ ((𝑇‘𝑥) ·ih 𝑥) ∧ ∀𝑥 ∈ ℋ 0 ≤ ((𝑈‘𝑥) ·ih 𝑥)))) |
30 | hmops 31541 | . . . 4 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 +op 𝑈) ∈ HrmOp) | |
31 | leoppos 31647 | . . . 4 ⊢ ((𝑇 +op 𝑈) ∈ HrmOp → ( 0hop ≤op (𝑇 +op 𝑈) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑇 +op 𝑈)‘𝑥) ·ih 𝑥))) | |
32 | 30, 31 | syl 17 | . . 3 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ( 0hop ≤op (𝑇 +op 𝑈) ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑇 +op 𝑈)‘𝑥) ·ih 𝑥))) |
33 | 26, 29, 32 | 3imtr4d 294 | . 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (( 0hop ≤op 𝑇 ∧ 0hop ≤op 𝑈) → 0hop ≤op (𝑇 +op 𝑈))) |
34 | 33 | imp 406 | 1 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ ( 0hop ≤op 𝑇 ∧ 0hop ≤op 𝑈)) → 0hop ≤op (𝑇 +op 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3060 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ℝcr 11113 0cc0 11114 + caddc 11117 ≤ cle 11254 ℋchba 30440 +ℎ cva 30441 ·ih csp 30443 +op chos 30459 0hop ch0o 30464 HrmOpcho 30471 ≤op cleo 30479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cc 10434 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194 ax-hilex 30520 ax-hfvadd 30521 ax-hvcom 30522 ax-hvass 30523 ax-hv0cl 30524 ax-hvaddid 30525 ax-hfvmul 30526 ax-hvmulid 30527 ax-hvmulass 30528 ax-hvdistr1 30529 ax-hvdistr2 30530 ax-hvmul0 30531 ax-hfi 30600 ax-his1 30603 ax-his2 30604 ax-his3 30605 ax-his4 30606 ax-hcompl 30723 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-omul 8475 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-acn 9941 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-rlim 15438 df-sum 15638 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-cn 22952 df-cnp 22953 df-lm 22954 df-haus 23040 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cfil 25004 df-cau 25005 df-cmet 25006 df-grpo 30014 df-gid 30015 df-ginv 30016 df-gdiv 30017 df-ablo 30066 df-vc 30080 df-nv 30113 df-va 30116 df-ba 30117 df-sm 30118 df-0v 30119 df-vs 30120 df-nmcv 30121 df-ims 30122 df-dip 30222 df-ssp 30243 df-ph 30334 df-cbn 30384 df-hnorm 30489 df-hba 30490 df-hvsub 30492 df-hlim 30493 df-hcau 30494 df-sh 30728 df-ch 30742 df-oc 30773 df-ch0 30774 df-shs 30829 df-pjh 30916 df-hosum 31251 df-homul 31252 df-hodif 31253 df-h0op 31269 df-hmop 31365 df-leop 31373 |
This theorem is referenced by: opsqrlem6 31666 |
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