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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.) (Revised by AV, 1-Nov-2024.) |
| 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 (𝐸‘ndx) |
| opsrbaslem.2 | ⊢ (𝐸‘ndx) ≠ (le‘ndx) |
| Ref | Expression |
|---|---|
| opsrbaslem | ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opsrbaslem.1 | . . . 4 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
| 2 | opsrbaslem.2 | . . . 4 ⊢ (𝐸‘ndx) ≠ (le‘ndx) | |
| 3 | 1, 2 | setsnid 17245 | . . 3 ⊢ (𝐸‘𝑆) = (𝐸‘(𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉)) |
| 4 | opsrbas.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
| 5 | opsrbas.o | . . . . 5 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) | |
| 6 | eqid 2737 | . . . . 5 ⊢ (le‘𝑂) = (le‘𝑂) | |
| 7 | simprl 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝐼 ∈ V) | |
| 8 | simprr 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑅 ∈ V) | |
| 9 | opsrbas.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
| 10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑇 ⊆ (𝐼 × 𝐼)) |
| 11 | 4, 5, 6, 7, 8, 10 | opsrval2 22066 | . . . 4 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑂 = (𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉)) |
| 12 | 11 | fveq2d 6910 | . . 3 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑂) = (𝐸‘(𝑆 sSet 〈(le‘ndx), (le‘𝑂)〉))) |
| 13 | 3, 12 | eqtr4id 2796 | . 2 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
| 14 | 0fv 6950 | . . . . . . 7 ⊢ (∅‘𝑇) = ∅ | |
| 15 | 14 | eqcomi 2746 | . . . . . 6 ⊢ ∅ = (∅‘𝑇) |
| 16 | reldmpsr 21934 | . . . . . . 7 ⊢ Rel dom mPwSer | |
| 17 | 16 | ovprc 7469 | . . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅) |
| 18 | reldmopsr 22063 | . . . . . . . 8 ⊢ Rel dom ordPwSer | |
| 19 | 18 | ovprc 7469 | . . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 ordPwSer 𝑅) = ∅) |
| 20 | 19 | fveq1d 6908 | . . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 ordPwSer 𝑅)‘𝑇) = (∅‘𝑇)) |
| 21 | 15, 17, 20 | 3eqtr4a 2803 | . . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ((𝐼 ordPwSer 𝑅)‘𝑇)) |
| 22 | 21, 4, 5 | 3eqtr4g 2802 | . . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = 𝑂) |
| 23 | 22 | fveq2d 6910 | . . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
| 24 | 23 | adantl 481 | . 2 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐸‘𝑆) = (𝐸‘𝑂)) |
| 25 | 13, 24 | pm2.61dan 813 | 1 ⊢ (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 〈cop 4632 × cxp 5683 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 Slot cslot 17218 ndxcnx 17230 lecple 17304 mPwSer cmps 21924 ordPwSer copws 21928 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-dec 12734 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ple 17317 df-psr 21929 df-opsr 21933 |
| This theorem is referenced by: opsrbas 22069 opsrplusg 22071 opsrmulr 22073 opsrvsca 22075 opsrsca 22077 |
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