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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 2734 | . 2 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
7 | eqid 2734 | . 2 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
8 | eqid 2734 | . 2 ⊢ (lt‘𝑅) = (lt‘𝑅) | |
9 | eqid 2734 | . 2 ⊢ (𝑇 <bag 𝐼) = (𝑇 <bag 𝐼) | |
10 | eqid 2734 | . 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 2734 | . 2 ⊢ (le‘𝑂) = (le‘𝑂) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | opsrtoslem2 22097 | 1 ⊢ (𝜑 → 𝑂 ∈ Toset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ∀wral 3063 ∃wrex 3072 {crab 3438 ⊆ wss 3970 class class class wbr 5169 We wwe 5653 × cxp 5697 ◡ccnv 5698 “ cima 5702 ‘cfv 6572 (class class class)co 7445 ↑m cmap 8880 Fincfn 8999 ℕcn 12289 ℕ0cn0 12549 Basecbs 17253 lecple 17313 ltcplt 18373 Tosetctos 18481 mPwSer cmps 21941 <bag cltb 21944 ordPwSer copws 21945 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-seqom 8500 df-1o 8518 df-2o 8519 df-oadd 8522 df-omul 8523 df-oexp 8524 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-oi 9575 df-cnf 9727 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-xnn0 12622 df-z 12636 df-dec 12755 df-uz 12900 df-fz 13564 df-hash 14376 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-tset 17325 df-ple 17326 df-proset 18360 df-poset 18378 df-plt 18395 df-toset 18482 df-psr 21946 df-ltbag 21949 df-opsr 21950 |
This theorem is referenced by: opsrso 22099 psr1tos 22204 |
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