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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 2823 | . 2 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
7 | eqid 2823 | . 2 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
8 | eqid 2823 | . 2 ⊢ (lt‘𝑅) = (lt‘𝑅) | |
9 | eqid 2823 | . 2 ⊢ (𝑇 <bag 𝐼) = (𝑇 <bag 𝐼) | |
10 | eqid 2823 | . 2 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
11 | biid 263 | . 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 2823 | . 2 ⊢ (le‘𝑂) = (le‘𝑂) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | opsrtoslem2 20267 | 1 ⊢ (𝜑 → 𝑂 ∈ Toset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 {crab 3144 ⊆ wss 3938 class class class wbr 5068 We wwe 5515 × cxp 5555 ◡ccnv 5556 “ cima 5560 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 Fincfn 8511 ℕcn 11640 ℕ0cn0 11900 Basecbs 16485 lecple 16574 ltcplt 17553 Tosetctos 17645 mPwSer cmps 20133 <bag cltb 20136 ordPwSer copws 20137 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-seqom 8086 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-oexp 8110 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-cnf 9127 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-tset 16586 df-ple 16587 df-proset 17540 df-poset 17558 df-plt 17570 df-toset 17646 df-psr 20138 df-ltbag 20141 df-opsr 20142 |
This theorem is referenced by: opsrso 20269 psr1tos 20359 |
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