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Mirrors > Home > MPE Home > Th. List > opsrbaslemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of opsrbaslem 22085 as of 1-Nov-2024. 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.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
opsrbas.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
opsrbas.o | ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) |
opsrbas.t | ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) |
opsrbaslemOLD.1 | ⊢ 𝐸 = Slot 𝑁 |
opsrbaslemOLD.2 | ⊢ 𝑁 ∈ ℕ |
opsrbaslemOLD.3 | ⊢ 𝑁 < ;10 |
Ref | Expression |
---|---|
opsrbaslemOLD | ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opsrbaslemOLD.1 | . . . . 5 ⊢ 𝐸 = Slot 𝑁 | |
2 | opsrbaslemOLD.2 | . . . . 5 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxid 17231 | . . . 4 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 2 | nnrei 12273 | . . . . . 6 ⊢ 𝑁 ∈ ℝ |
5 | opsrbaslemOLD.3 | . . . . . 6 ⊢ 𝑁 < ;10 | |
6 | 4, 5 | ltneii 11372 | . . . . 5 ⊢ 𝑁 ≠ ;10 |
7 | 1, 2 | ndxarg 17230 | . . . . . 6 ⊢ (𝐸‘ndx) = 𝑁 |
8 | plendx 17412 | . . . . . 6 ⊢ (le‘ndx) = ;10 | |
9 | 7, 8 | neeq12i 3005 | . . . . 5 ⊢ ((𝐸‘ndx) ≠ (le‘ndx) ↔ 𝑁 ≠ ;10) |
10 | 6, 9 | mpbir 231 | . . . 4 ⊢ (𝐸‘ndx) ≠ (le‘ndx) |
11 | 3, 10 | setsnid 17243 | . . 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 2735 | . . . . 5 ⊢ (le‘𝑂) = (le‘𝑂) | |
15 | simprl 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝐼 ∈ V) | |
16 | simprr 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑅 ∈ V) | |
17 | opsrbas.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
18 | 17 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑇 ⊆ (𝐼 × 𝐼)) |
19 | 12, 13, 14, 15, 16, 18 | opsrval2 22084 | . . . 4 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑂 = (𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉)) |
20 | 19 | fveq2d 6911 | . . 3 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑂) = (𝐸‘(𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉))) |
21 | 11, 20 | eqtr4id 2794 | . 2 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
22 | 0fv 6951 | . . . . . . 7 ⊢ (∅‘𝑇) = ∅ | |
23 | 22 | eqcomi 2744 | . . . . . 6 ⊢ ∅ = (∅‘𝑇) |
24 | reldmpsr 21952 | . . . . . . 7 ⊢ Rel dom mPwSer | |
25 | 24 | ovprc 7469 | . . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅) |
26 | reldmopsr 22081 | . . . . . . . 8 ⊢ Rel dom ordPwSer | |
27 | 26 | ovprc 7469 | . . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 ordPwSer 𝑅) = ∅) |
28 | 27 | fveq1d 6909 | . . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 ordPwSer 𝑅)‘𝑇) = (∅‘𝑇)) |
29 | 23, 25, 28 | 3eqtr4a 2801 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ((𝐼 ordPwSer 𝑅)‘𝑇)) |
30 | 29 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐼 mPwSer 𝑅) = ((𝐼 ordPwSer 𝑅)‘𝑇)) |
31 | 30, 12, 13 | 3eqtr4g 2800 | . . 3 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑆 = 𝑂) |
32 | 31 | fveq2d 6911 | . 2 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
33 | 21, 32 | pm2.61dan 813 | 1 ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 〈cop 4637 class class class wbr 5148 × cxp 5687 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 < clt 11293 ℕcn 12264 ;cdc 12731 sSet csts 17197 Slot cslot 17215 ndxcnx 17227 lecple 17305 mPwSer cmps 21942 ordPwSer copws 21946 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-dec 12732 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ple 17318 df-psr 21947 df-opsr 21951 |
This theorem is referenced by: opsrbasOLD 22088 opsrplusgOLD 22090 opsrmulrOLD 22092 opsrvscaOLD 22094 opsrscaOLD 22096 |
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