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Mirrors > Home > MPE Home > Th. List > opsrval2 | Structured version Visualization version GIF version |
Description: Self-referential expression for the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) |
Ref | Expression |
---|---|
opsrval2.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
opsrval2.o | ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) |
opsrval2.l | ⊢ ≤ = (le‘𝑂) |
opsrval2.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
opsrval2.r | ⊢ (𝜑 → 𝑅 ∈ 𝑊) |
opsrval2.t | ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) |
Ref | Expression |
---|---|
opsrval2 | ⊢ (𝜑 → 𝑂 = (𝑆 sSet 〈(le‘ndx), ≤ 〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opsrval2.s | . . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
2 | opsrval2.o | . . 3 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) | |
3 | eqid 2728 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
4 | eqid 2728 | . . 3 ⊢ (lt‘𝑅) = (lt‘𝑅) | |
5 | eqid 2728 | . . 3 ⊢ (𝑇 <bag 𝐼) = (𝑇 <bag 𝐼) | |
6 | eqid 2728 | . . 3 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
7 | eqid 2728 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑆) ∧ (∃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))) ∨ 𝑥 = 𝑦))} = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑆) ∧ (∃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))) ∨ 𝑥 = 𝑦))} | |
8 | opsrval2.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
9 | opsrval2.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊) | |
10 | opsrval2.t | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | opsrval 21978 | . 2 ⊢ (𝜑 → 𝑂 = (𝑆 sSet 〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑆) ∧ (∃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))) ∨ 𝑥 = 𝑦))}〉)) |
12 | opsrval2.l | . . . . 5 ⊢ ≤ = (le‘𝑂) | |
13 | 1, 2, 3, 4, 5, 6, 12, 10 | opsrle 21979 | . . . 4 ⊢ (𝜑 → ≤ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑆) ∧ (∃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))) ∨ 𝑥 = 𝑦))}) |
14 | 13 | opeq2d 4877 | . . 3 ⊢ (𝜑 → 〈(le‘ndx), ≤ 〉 = 〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑆) ∧ (∃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))) ∨ 𝑥 = 𝑦))}〉) |
15 | 14 | oveq2d 7431 | . 2 ⊢ (𝜑 → (𝑆 sSet 〈(le‘ndx), ≤ 〉) = (𝑆 sSet 〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑆) ∧ (∃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))) ∨ 𝑥 = 𝑦))}〉)) |
16 | 11, 15 | eqtr4d 2771 | 1 ⊢ (𝜑 → 𝑂 = (𝑆 sSet 〈(le‘ndx), ≤ 〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 {crab 3428 ⊆ wss 3945 {cpr 4627 〈cop 4631 class class class wbr 5143 {copab 5205 × cxp 5671 ◡ccnv 5672 “ cima 5676 ‘cfv 6543 (class class class)co 7415 ↑m cmap 8839 Fincfn 8958 ℕcn 12237 ℕ0cn0 12497 sSet csts 17126 ndxcnx 17156 Basecbs 17174 lecple 17234 ltcplt 18294 mPwSer cmps 21831 <bag cltb 21834 ordPwSer copws 21835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-ltxr 11278 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-dec 12703 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ple 17247 df-psr 21836 df-opsr 21840 |
This theorem is referenced by: opsrbaslem 21981 opsrbaslemOLD 21982 |
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