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Mirrors > Home > MPE Home > Th. List > opsrbaslem | Structured version Visualization version GIF version |
Description: Get a component of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 9-Sep-2021.) |
Ref | Expression |
---|---|
opsrbas.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
opsrbas.o | ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) |
opsrbas.t | ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) |
opsrbaslem.1 | ⊢ 𝐸 = Slot 𝑁 |
opsrbaslem.2 | ⊢ 𝑁 ∈ ℕ |
opsrbaslem.3 | ⊢ 𝑁 < ;10 |
Ref | Expression |
---|---|
opsrbaslem | ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opsrbas.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
2 | opsrbas.o | . . . . 5 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) | |
3 | eqid 2622 | . . . . 5 ⊢ (le‘𝑂) = (le‘𝑂) | |
4 | simprl 794 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝐼 ∈ V) | |
5 | simprr 796 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑅 ∈ V) | |
6 | opsrbas.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
7 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑇 ⊆ (𝐼 × 𝐼)) |
8 | 1, 2, 3, 4, 5, 7 | opsrval2 19476 | . . . 4 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑂 = (𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉)) |
9 | 8 | fveq2d 6195 | . . 3 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑂) = (𝐸‘(𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉))) |
10 | opsrbaslem.1 | . . . . 5 ⊢ 𝐸 = Slot 𝑁 | |
11 | opsrbaslem.2 | . . . . 5 ⊢ 𝑁 ∈ ℕ | |
12 | 10, 11 | ndxid 15883 | . . . 4 ⊢ 𝐸 = Slot (𝐸‘ndx) |
13 | 11 | nnrei 11029 | . . . . . 6 ⊢ 𝑁 ∈ ℝ |
14 | opsrbaslem.3 | . . . . . 6 ⊢ 𝑁 < ;10 | |
15 | 13, 14 | ltneii 10150 | . . . . 5 ⊢ 𝑁 ≠ ;10 |
16 | 10, 11 | ndxarg 15882 | . . . . . 6 ⊢ (𝐸‘ndx) = 𝑁 |
17 | plendx 16047 | . . . . . 6 ⊢ (le‘ndx) = ;10 | |
18 | 16, 17 | neeq12i 2860 | . . . . 5 ⊢ ((𝐸‘ndx) ≠ (le‘ndx) ↔ 𝑁 ≠ ;10) |
19 | 15, 18 | mpbir 221 | . . . 4 ⊢ (𝐸‘ndx) ≠ (le‘ndx) |
20 | 12, 19 | setsnid 15915 | . . 3 ⊢ (𝐸‘𝑆) = (𝐸‘(𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉)) |
21 | 9, 20 | syl6reqr 2675 | . 2 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
22 | 0fv 6227 | . . . . . . 7 ⊢ (∅‘𝑇) = ∅ | |
23 | 22 | eqcomi 2631 | . . . . . 6 ⊢ ∅ = (∅‘𝑇) |
24 | reldmpsr 19361 | . . . . . . 7 ⊢ Rel dom mPwSer | |
25 | 24 | ovprc 6683 | . . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅) |
26 | reldmopsr 19473 | . . . . . . . 8 ⊢ Rel dom ordPwSer | |
27 | 26 | ovprc 6683 | . . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 ordPwSer 𝑅) = ∅) |
28 | 27 | fveq1d 6193 | . . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 ordPwSer 𝑅)‘𝑇) = (∅‘𝑇)) |
29 | 23, 25, 28 | 3eqtr4a 2682 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ((𝐼 ordPwSer 𝑅)‘𝑇)) |
30 | 29 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐼 mPwSer 𝑅) = ((𝐼 ordPwSer 𝑅)‘𝑇)) |
31 | 30, 1, 2 | 3eqtr4g 2681 | . . 3 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑆 = 𝑂) |
32 | 31 | fveq2d 6195 | . 2 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
33 | 21, 32 | pm2.61dan 832 | 1 ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 〈cop 4183 class class class wbr 4653 × cxp 5112 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 < clt 10074 ℕcn 11020 ;cdc 11493 ndxcnx 15854 sSet csts 15855 Slot cslot 15856 lecple 15948 mPwSer cmps 19351 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-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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-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-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-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-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 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-dec 11494 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ple 15961 df-psr 19356 df-opsr 19360 |
This theorem is referenced by: opsrbas 19479 opsrplusg 19480 opsrmulr 19481 opsrvsca 19482 opsrsca 19483 |
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