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Mirrors > Home > MPE Home > Th. List > opsr1 | Structured version Visualization version GIF version |
Description: One in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
opsr0.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
opsr0.o | ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) |
opsr0.t | ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) |
Ref | Expression |
---|---|
opsr1 | ⊢ (𝜑 → (1r‘𝑆) = (1r‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2739 | . 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆)) | |
2 | opsr0.s | . . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
3 | opsr0.o | . . 3 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) | |
4 | opsr0.t | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) | |
5 | 2, 3, 4 | opsrbas 21374 | . 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑂)) |
6 | 2, 3, 4 | opsrmulr 21378 | . . 3 ⊢ (𝜑 → (.r‘𝑆) = (.r‘𝑂)) |
7 | 6 | oveqdr 7378 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(.r‘𝑆)𝑦) = (𝑥(.r‘𝑂)𝑦)) |
8 | 1, 5, 7 | rngidpropd 20047 | 1 ⊢ (𝜑 → (1r‘𝑆) = (1r‘𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3909 × cxp 5629 ‘cfv 6492 (class class class)co 7350 Basecbs 17018 .rcmulr 17069 1rcur 19842 mPwSer cmps 21229 ordPwSer copws 21233 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-dec 12552 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-plusg 17081 df-mulr 17082 df-ple 17088 df-0g 17258 df-mgp 19826 df-ur 19843 df-psr 21234 df-opsr 21238 |
This theorem is referenced by: (None) |
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