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| Description: The derivative of one is zero. (Contributed by SN, 25-Apr-2025.) | 
| Ref | Expression | 
|---|---|
| psd1.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | 
| psd1.u | ⊢ 1 = (1r‘𝑆) | 
| psd1.z | ⊢ 0 = (0g‘𝑆) | 
| psd1.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) | 
| psd1.r | ⊢ (𝜑 → 𝑅 ∈ CRing) | 
| psd1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐼) | 
| Ref | Expression | 
|---|---|
| psd1 | ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 ) = 0 ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | psd1.s | . . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
| 2 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
| 3 | eqid 2737 | . . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
| 4 | eqid 2737 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
| 5 | psd1.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
| 6 | psd1.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
| 7 | psd1.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
| 8 | 1, 7, 5 | psrcrng 21992 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ CRing) | 
| 9 | 8 | crngringd 20243 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring) | 
| 10 | psd1.u | . . . . . . 7 ⊢ 1 = (1r‘𝑆) | |
| 11 | 2, 10 | ringidcl 20262 | . . . . . 6 ⊢ (𝑆 ∈ Ring → 1 ∈ (Base‘𝑆)) | 
| 12 | 9, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 1 ∈ (Base‘𝑆)) | 
| 13 | 1, 2, 3, 4, 5, 6, 12, 12 | psdmul 22170 | . . . 4 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘( 1 (.r‘𝑆) 1 )) = (((((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )(.r‘𝑆) 1 )(+g‘𝑆)( 1 (.r‘𝑆)(((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )))) | 
| 14 | 2, 4, 10, 9, 12 | ringlidmd 20269 | . . . . 5 ⊢ (𝜑 → ( 1 (.r‘𝑆) 1 ) = 1 ) | 
| 15 | 14 | fveq2d 6910 | . . . 4 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘( 1 (.r‘𝑆) 1 )) = (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )) | 
| 16 | 5 | crnggrpd 20244 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Grp) | 
| 17 | 16 | grpmgmd 18979 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Mgm) | 
| 18 | 1, 2, 17, 6, 12 | psdcl 22165 | . . . . . 6 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 ) ∈ (Base‘𝑆)) | 
| 19 | 2, 4, 10, 9, 18 | ringridmd 20270 | . . . . 5 ⊢ (𝜑 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )(.r‘𝑆) 1 ) = (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )) | 
| 20 | 2, 4, 10, 9, 18 | ringlidmd 20269 | . . . . 5 ⊢ (𝜑 → ( 1 (.r‘𝑆)(((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )) = (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )) | 
| 21 | 19, 20 | oveq12d 7449 | . . . 4 ⊢ (𝜑 → (((((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )(.r‘𝑆) 1 )(+g‘𝑆)( 1 (.r‘𝑆)(((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 ))) = ((((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )(+g‘𝑆)(((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 ))) | 
| 22 | 13, 15, 21 | 3eqtr3rd 2786 | . . 3 ⊢ (𝜑 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )(+g‘𝑆)(((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )) = (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )) | 
| 23 | 8 | crnggrpd 20244 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Grp) | 
| 24 | psd1.z | . . . . 5 ⊢ 0 = (0g‘𝑆) | |
| 25 | 2, 3, 24 | grpid 18993 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 ) ∈ (Base‘𝑆)) → (((((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )(+g‘𝑆)(((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )) = (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 ) ↔ 0 = (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 ))) | 
| 26 | 23, 18, 25 | syl2anc 584 | . . 3 ⊢ (𝜑 → (((((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )(+g‘𝑆)(((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )) = (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 ) ↔ 0 = (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 ))) | 
| 27 | 22, 26 | mpbid 232 | . 2 ⊢ (𝜑 → 0 = (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 )) | 
| 28 | 27 | eqcomd 2743 | 1 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 ) = 0 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 0gc0g 17484 Grpcgrp 18951 1rcur 20178 Ringcrg 20230 CRingccrg 20231 mPwSer cmps 21924 mPSDer cpsd 22134 | 
| 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 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 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-rmo 3380 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-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 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-se 5638 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-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 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-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-mulg 19086 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-psr 21929 df-psd 22160 | 
| This theorem is referenced by: psdascl 22172 | 
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