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Mirrors > Home > MPE Home > Th. List > opsrso | Structured version Visualization version GIF version |
Description: The ordered power series structure is a totally ordered set. (Contributed by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
opsrso.o | ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) |
opsrso.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
opsrso.r | ⊢ (𝜑 → 𝑅 ∈ Toset) |
opsrso.t | ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) |
opsrso.w | ⊢ (𝜑 → 𝑇 We 𝐼) |
opsrso.l | ⊢ ≤ = (lt‘𝑂) |
opsrso.b | ⊢ 𝐵 = (Base‘𝑂) |
Ref | Expression |
---|---|
opsrso | ⊢ (𝜑 → ≤ Or 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opsrso.o | . . . 4 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) | |
2 | opsrso.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
3 | opsrso.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Toset) | |
4 | opsrso.t | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
5 | opsrso.w | . . . 4 ⊢ (𝜑 → 𝑇 We 𝐼) | |
6 | 1, 2, 3, 4, 5 | opsrtos 19486 | . . 3 ⊢ (𝜑 → 𝑂 ∈ Toset) |
7 | opsrso.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑂) | |
8 | eqid 2622 | . . . . 5 ⊢ (le‘𝑂) = (le‘𝑂) | |
9 | opsrso.l | . . . . 5 ⊢ ≤ = (lt‘𝑂) | |
10 | 7, 8, 9 | tosso 17036 | . . . 4 ⊢ (𝑂 ∈ Toset → (𝑂 ∈ Toset ↔ ( ≤ Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝑂)))) |
11 | 10 | ibi 256 | . . 3 ⊢ (𝑂 ∈ Toset → ( ≤ Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝑂))) |
12 | 6, 11 | syl 17 | . 2 ⊢ (𝜑 → ( ≤ Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝑂))) |
13 | 12 | simpld 475 | 1 ⊢ (𝜑 → ≤ Or 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 I cid 5023 Or wor 5034 We wwe 5072 × cxp 5112 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 ltcplt 16941 Tosetctos 17033 ordPwSer copws 19355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seqom 7543 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-oexp 7566 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-cnf 8559 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-tset 15960 df-ple 15961 df-preset 16928 df-poset 16946 df-plt 16958 df-toset 17034 df-psr 19356 df-ltbag 19359 df-opsr 19360 |
This theorem is referenced by: (None) |
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