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Mirrors > Home > HSE Home > Th. List > stlei | Structured version Visualization version GIF version |
Description: Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
stle.1 | ⊢ 𝐴 ∈ Cℋ |
stle.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
stlei | ⊢ (𝑆 ∈ States → (𝐴 ⊆ 𝐵 → (𝑆‘𝐴) ≤ (𝑆‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stle.2 | . . . . . . . . . 10 ⊢ 𝐵 ∈ Cℋ | |
2 | 1 | chshii 30975 | . . . . . . . . 9 ⊢ 𝐵 ∈ Sℋ |
3 | shococss 31042 | . . . . . . . . 9 ⊢ (𝐵 ∈ Sℋ → 𝐵 ⊆ (⊥‘(⊥‘𝐵))) | |
4 | 2, 3 | ax-mp 5 | . . . . . . . 8 ⊢ 𝐵 ⊆ (⊥‘(⊥‘𝐵)) |
5 | sstr2 3982 | . . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (⊥‘(⊥‘𝐵)) → 𝐴 ⊆ (⊥‘(⊥‘𝐵)))) | |
6 | 4, 5 | mpi 20 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (⊥‘(⊥‘𝐵))) |
7 | stle.1 | . . . . . . . 8 ⊢ 𝐴 ∈ Cℋ | |
8 | 1 | choccli 31055 | . . . . . . . 8 ⊢ (⊥‘𝐵) ∈ Cℋ |
9 | 7, 8 | pm3.2i 470 | . . . . . . 7 ⊢ (𝐴 ∈ Cℋ ∧ (⊥‘𝐵) ∈ Cℋ ) |
10 | 6, 9 | jctil 519 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ((𝐴 ∈ Cℋ ∧ (⊥‘𝐵) ∈ Cℋ ) ∧ 𝐴 ⊆ (⊥‘(⊥‘𝐵)))) |
11 | stj 31983 | . . . . . 6 ⊢ (𝑆 ∈ States → (((𝐴 ∈ Cℋ ∧ (⊥‘𝐵) ∈ Cℋ ) ∧ 𝐴 ⊆ (⊥‘(⊥‘𝐵))) → (𝑆‘(𝐴 ∨ℋ (⊥‘𝐵))) = ((𝑆‘𝐴) + (𝑆‘(⊥‘𝐵))))) | |
12 | 10, 11 | syl5 34 | . . . . 5 ⊢ (𝑆 ∈ States → (𝐴 ⊆ 𝐵 → (𝑆‘(𝐴 ∨ℋ (⊥‘𝐵))) = ((𝑆‘𝐴) + (𝑆‘(⊥‘𝐵))))) |
13 | 12 | imp 406 | . . . 4 ⊢ ((𝑆 ∈ States ∧ 𝐴 ⊆ 𝐵) → (𝑆‘(𝐴 ∨ℋ (⊥‘𝐵))) = ((𝑆‘𝐴) + (𝑆‘(⊥‘𝐵)))) |
14 | 7, 8 | chjcli 31205 | . . . . . . 7 ⊢ (𝐴 ∨ℋ (⊥‘𝐵)) ∈ Cℋ |
15 | stle1 31973 | . . . . . . 7 ⊢ (𝑆 ∈ States → ((𝐴 ∨ℋ (⊥‘𝐵)) ∈ Cℋ → (𝑆‘(𝐴 ∨ℋ (⊥‘𝐵))) ≤ 1)) | |
16 | 14, 15 | mpi 20 | . . . . . 6 ⊢ (𝑆 ∈ States → (𝑆‘(𝐴 ∨ℋ (⊥‘𝐵))) ≤ 1) |
17 | 1 | sto1i 31984 | . . . . . 6 ⊢ (𝑆 ∈ States → ((𝑆‘𝐵) + (𝑆‘(⊥‘𝐵))) = 1) |
18 | 16, 17 | breqtrrd 5167 | . . . . 5 ⊢ (𝑆 ∈ States → (𝑆‘(𝐴 ∨ℋ (⊥‘𝐵))) ≤ ((𝑆‘𝐵) + (𝑆‘(⊥‘𝐵)))) |
19 | 18 | adantr 480 | . . . 4 ⊢ ((𝑆 ∈ States ∧ 𝐴 ⊆ 𝐵) → (𝑆‘(𝐴 ∨ℋ (⊥‘𝐵))) ≤ ((𝑆‘𝐵) + (𝑆‘(⊥‘𝐵)))) |
20 | 13, 19 | eqbrtrrd 5163 | . . 3 ⊢ ((𝑆 ∈ States ∧ 𝐴 ⊆ 𝐵) → ((𝑆‘𝐴) + (𝑆‘(⊥‘𝐵))) ≤ ((𝑆‘𝐵) + (𝑆‘(⊥‘𝐵)))) |
21 | stcl 31964 | . . . . . . 7 ⊢ (𝑆 ∈ States → (𝐴 ∈ Cℋ → (𝑆‘𝐴) ∈ ℝ)) | |
22 | 7, 21 | mpi 20 | . . . . . 6 ⊢ (𝑆 ∈ States → (𝑆‘𝐴) ∈ ℝ) |
23 | stcl 31964 | . . . . . . 7 ⊢ (𝑆 ∈ States → (𝐵 ∈ Cℋ → (𝑆‘𝐵) ∈ ℝ)) | |
24 | 1, 23 | mpi 20 | . . . . . 6 ⊢ (𝑆 ∈ States → (𝑆‘𝐵) ∈ ℝ) |
25 | stcl 31964 | . . . . . . 7 ⊢ (𝑆 ∈ States → ((⊥‘𝐵) ∈ Cℋ → (𝑆‘(⊥‘𝐵)) ∈ ℝ)) | |
26 | 8, 25 | mpi 20 | . . . . . 6 ⊢ (𝑆 ∈ States → (𝑆‘(⊥‘𝐵)) ∈ ℝ) |
27 | 22, 24, 26 | 3jca 1125 | . . . . 5 ⊢ (𝑆 ∈ States → ((𝑆‘𝐴) ∈ ℝ ∧ (𝑆‘𝐵) ∈ ℝ ∧ (𝑆‘(⊥‘𝐵)) ∈ ℝ)) |
28 | 27 | adantr 480 | . . . 4 ⊢ ((𝑆 ∈ States ∧ 𝐴 ⊆ 𝐵) → ((𝑆‘𝐴) ∈ ℝ ∧ (𝑆‘𝐵) ∈ ℝ ∧ (𝑆‘(⊥‘𝐵)) ∈ ℝ)) |
29 | leadd1 11681 | . . . 4 ⊢ (((𝑆‘𝐴) ∈ ℝ ∧ (𝑆‘𝐵) ∈ ℝ ∧ (𝑆‘(⊥‘𝐵)) ∈ ℝ) → ((𝑆‘𝐴) ≤ (𝑆‘𝐵) ↔ ((𝑆‘𝐴) + (𝑆‘(⊥‘𝐵))) ≤ ((𝑆‘𝐵) + (𝑆‘(⊥‘𝐵))))) | |
30 | 28, 29 | syl 17 | . . 3 ⊢ ((𝑆 ∈ States ∧ 𝐴 ⊆ 𝐵) → ((𝑆‘𝐴) ≤ (𝑆‘𝐵) ↔ ((𝑆‘𝐴) + (𝑆‘(⊥‘𝐵))) ≤ ((𝑆‘𝐵) + (𝑆‘(⊥‘𝐵))))) |
31 | 20, 30 | mpbird 257 | . 2 ⊢ ((𝑆 ∈ States ∧ 𝐴 ⊆ 𝐵) → (𝑆‘𝐴) ≤ (𝑆‘𝐵)) |
32 | 31 | ex 412 | 1 ⊢ (𝑆 ∈ States → (𝐴 ⊆ 𝐵 → (𝑆‘𝐴) ≤ (𝑆‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3941 class class class wbr 5139 ‘cfv 6534 (class class class)co 7402 ℝcr 11106 1c1 11108 + caddc 11110 ≤ cle 11248 Sℋ csh 30676 Cℋ cch 30677 ⊥cort 30678 ∨ℋ chj 30681 Statescst 30710 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 ax-hilex 30747 ax-hfvadd 30748 ax-hvcom 30749 ax-hvass 30750 ax-hv0cl 30751 ax-hvaddid 30752 ax-hfvmul 30753 ax-hvmulid 30754 ax-hvmulass 30755 ax-hvdistr1 30756 ax-hvdistr2 30757 ax-hvmul0 30758 ax-hfi 30827 ax-his1 30830 ax-his2 30831 ax-his3 30832 ax-his4 30833 ax-hcompl 30950 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ioo 13329 df-icc 13332 df-fz 13486 df-fzo 13629 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 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 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-mulg 18992 df-cntz 19229 df-cmn 19698 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-cnfld 21235 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-cn 23075 df-cnp 23076 df-lm 23077 df-haus 23163 df-tx 23410 df-hmeo 23603 df-xms 24170 df-ms 24171 df-tms 24172 df-cau 25128 df-grpo 30241 df-gid 30242 df-ginv 30243 df-gdiv 30244 df-ablo 30293 df-vc 30307 df-nv 30340 df-va 30343 df-ba 30344 df-sm 30345 df-0v 30346 df-vs 30347 df-nmcv 30348 df-ims 30349 df-dip 30449 df-hnorm 30716 df-hvsub 30719 df-hlim 30720 df-hcau 30721 df-sh 30955 df-ch 30969 df-oc 31000 df-ch0 31001 df-chj 31058 df-st 31959 |
This theorem is referenced by: stlesi 31989 stm1i 31991 |
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