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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 | opsrbaslem.1 | . . . . 5 ⊢ 𝐸 = Slot 𝑁 | |
2 | opsrbaslem.2 | . . . . 5 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxid 16501 | . . . 4 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 2 | nnrei 11634 | . . . . . 6 ⊢ 𝑁 ∈ ℝ |
5 | opsrbaslem.3 | . . . . . 6 ⊢ 𝑁 < ;10 | |
6 | 4, 5 | ltneii 10742 | . . . . 5 ⊢ 𝑁 ≠ ;10 |
7 | 1, 2 | ndxarg 16500 | . . . . . 6 ⊢ (𝐸‘ndx) = 𝑁 |
8 | plendx 16658 | . . . . . 6 ⊢ (le‘ndx) = ;10 | |
9 | 7, 8 | neeq12i 3053 | . . . . 5 ⊢ ((𝐸‘ndx) ≠ (le‘ndx) ↔ 𝑁 ≠ ;10) |
10 | 6, 9 | mpbir 234 | . . . 4 ⊢ (𝐸‘ndx) ≠ (le‘ndx) |
11 | 3, 10 | setsnid 16531 | . . 3 ⊢ (𝐸‘𝑆) = (𝐸‘(𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉)) |
12 | opsrbas.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
13 | opsrbas.o | . . . . 5 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) | |
14 | eqid 2798 | . . . . 5 ⊢ (le‘𝑂) = (le‘𝑂) | |
15 | simprl 770 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝐼 ∈ V) | |
16 | simprr 772 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑅 ∈ V) | |
17 | opsrbas.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
18 | 17 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑇 ⊆ (𝐼 × 𝐼)) |
19 | 12, 13, 14, 15, 16, 18 | opsrval2 20716 | . . . 4 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑂 = (𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉)) |
20 | 19 | fveq2d 6649 | . . 3 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑂) = (𝐸‘(𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉))) |
21 | 11, 20 | eqtr4id 2852 | . 2 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
22 | 0fv 6684 | . . . . . . 7 ⊢ (∅‘𝑇) = ∅ | |
23 | 22 | eqcomi 2807 | . . . . . 6 ⊢ ∅ = (∅‘𝑇) |
24 | reldmpsr 20599 | . . . . . . 7 ⊢ Rel dom mPwSer | |
25 | 24 | ovprc 7173 | . . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅) |
26 | reldmopsr 20713 | . . . . . . . 8 ⊢ Rel dom ordPwSer | |
27 | 26 | ovprc 7173 | . . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 ordPwSer 𝑅) = ∅) |
28 | 27 | fveq1d 6647 | . . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 ordPwSer 𝑅)‘𝑇) = (∅‘𝑇)) |
29 | 23, 25, 28 | 3eqtr4a 2859 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ((𝐼 ordPwSer 𝑅)‘𝑇)) |
30 | 29 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐼 mPwSer 𝑅) = ((𝐼 ordPwSer 𝑅)‘𝑇)) |
31 | 30, 12, 13 | 3eqtr4g 2858 | . . 3 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑆 = 𝑂) |
32 | 31 | fveq2d 6649 | . 2 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
33 | 21, 32 | pm2.61dan 812 | 1 ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 〈cop 4531 class class class wbr 5030 × cxp 5517 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 < clt 10664 ℕcn 11625 ;cdc 12086 ndxcnx 16472 sSet csts 16473 Slot cslot 16474 lecple 16564 mPwSer cmps 20589 ordPwSer copws 20593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-dec 12087 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ple 16577 df-psr 20594 df-opsr 20598 |
This theorem is referenced by: opsrbas 20718 opsrplusg 20719 opsrmulr 20720 opsrvsca 20721 opsrsca 20722 |
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