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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 2821 | . . . . 5 ⊢ (le‘𝑂) = (le‘𝑂) | |
4 | simprl 769 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝐼 ∈ V) | |
5 | simprr 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑅 ∈ V) | |
6 | opsrbas.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
7 | 6 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑇 ⊆ (𝐼 × 𝐼)) |
8 | 1, 2, 3, 4, 5, 7 | opsrval2 20251 | . . . 4 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑂 = (𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉)) |
9 | 8 | fveq2d 6668 | . . 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 16503 | . . . 4 ⊢ 𝐸 = Slot (𝐸‘ndx) |
13 | 11 | nnrei 11641 | . . . . . 6 ⊢ 𝑁 ∈ ℝ |
14 | opsrbaslem.3 | . . . . . 6 ⊢ 𝑁 < ;10 | |
15 | 13, 14 | ltneii 10747 | . . . . 5 ⊢ 𝑁 ≠ ;10 |
16 | 10, 11 | ndxarg 16502 | . . . . . 6 ⊢ (𝐸‘ndx) = 𝑁 |
17 | plendx 16660 | . . . . . 6 ⊢ (le‘ndx) = ;10 | |
18 | 16, 17 | neeq12i 3082 | . . . . 5 ⊢ ((𝐸‘ndx) ≠ (le‘ndx) ↔ 𝑁 ≠ ;10) |
19 | 15, 18 | mpbir 233 | . . . 4 ⊢ (𝐸‘ndx) ≠ (le‘ndx) |
20 | 12, 19 | setsnid 16533 | . . 3 ⊢ (𝐸‘𝑆) = (𝐸‘(𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉)) |
21 | 9, 20 | syl6reqr 2875 | . 2 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
22 | 0fv 6703 | . . . . . . 7 ⊢ (∅‘𝑇) = ∅ | |
23 | 22 | eqcomi 2830 | . . . . . 6 ⊢ ∅ = (∅‘𝑇) |
24 | reldmpsr 20135 | . . . . . . 7 ⊢ Rel dom mPwSer | |
25 | 24 | ovprc 7188 | . . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅) |
26 | reldmopsr 20248 | . . . . . . . 8 ⊢ Rel dom ordPwSer | |
27 | 26 | ovprc 7188 | . . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 ordPwSer 𝑅) = ∅) |
28 | 27 | fveq1d 6666 | . . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 ordPwSer 𝑅)‘𝑇) = (∅‘𝑇)) |
29 | 23, 25, 28 | 3eqtr4a 2882 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ((𝐼 ordPwSer 𝑅)‘𝑇)) |
30 | 29 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐼 mPwSer 𝑅) = ((𝐼 ordPwSer 𝑅)‘𝑇)) |
31 | 30, 1, 2 | 3eqtr4g 2881 | . . 3 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑆 = 𝑂) |
32 | 31 | fveq2d 6668 | . 2 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
33 | 21, 32 | pm2.61dan 811 | 1 ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 〈cop 4566 class class class wbr 5058 × cxp 5547 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 < clt 10669 ℕcn 11632 ;cdc 12092 ndxcnx 16474 sSet csts 16475 Slot cslot 16476 lecple 16566 mPwSer cmps 20125 ordPwSer copws 20129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-dec 12093 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ple 16579 df-psr 20130 df-opsr 20134 |
This theorem is referenced by: opsrbas 20253 opsrplusg 20254 opsrmulr 20255 opsrvsca 20256 opsrsca 20257 |
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