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Mirrors > Home > MPE Home > Th. List > opsrtos | 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) |
Ref | Expression |
---|---|
opsrtos | ⊢ (𝜑 → 𝑂 ∈ Toset) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opsrso.o | . 2 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) | |
2 | opsrso.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
3 | opsrso.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Toset) | |
4 | opsrso.t | . 2 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
5 | opsrso.w | . 2 ⊢ (𝜑 → 𝑇 We 𝐼) | |
6 | eqid 2740 | . 2 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
7 | eqid 2740 | . 2 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
8 | eqid 2740 | . 2 ⊢ (lt‘𝑅) = (lt‘𝑅) | |
9 | eqid 2740 | . 2 ⊢ (𝑇 <bag 𝐼) = (𝑇 <bag 𝐼) | |
10 | eqid 2740 | . 2 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
11 | biid 261 | . 2 ⊢ (∃𝑧 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ((𝑥‘𝑧)(lt‘𝑅)(𝑦‘𝑧) ∧ ∀𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} (𝑤(𝑇 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ∃𝑧 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ((𝑥‘𝑧)(lt‘𝑅)(𝑦‘𝑧) ∧ ∀𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} (𝑤(𝑇 <bag 𝐼)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))) | |
12 | eqid 2740 | . 2 ⊢ (le‘𝑂) = (le‘𝑂) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | opsrtoslem2 22105 | 1 ⊢ (𝜑 → 𝑂 ∈ Toset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 {crab 3443 ⊆ wss 3976 class class class wbr 5166 We wwe 5651 × cxp 5698 ◡ccnv 5699 “ cima 5703 ‘cfv 6575 (class class class)co 7450 ↑m cmap 8886 Fincfn 9005 ℕcn 12295 ℕ0cn0 12555 Basecbs 17260 lecple 17320 ltcplt 18380 Tosetctos 18488 mPwSer cmps 21949 <bag cltb 21952 ordPwSer copws 21953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-om 7906 df-1st 8032 df-2nd 8033 df-supp 8204 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-seqom 8506 df-1o 8524 df-2o 8525 df-oadd 8528 df-omul 8529 df-oexp 8530 df-er 8765 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-fsupp 9434 df-oi 9581 df-cnf 9733 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-xnn0 12628 df-z 12642 df-dec 12761 df-uz 12906 df-fz 13570 df-hash 14382 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-plusg 17326 df-mulr 17327 df-sca 17329 df-vsca 17330 df-tset 17332 df-ple 17333 df-proset 18367 df-poset 18385 df-plt 18402 df-toset 18489 df-psr 21954 df-ltbag 21957 df-opsr 21958 |
This theorem is referenced by: opsrso 22107 psr1tos 22213 |
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