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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmpsr1 | Structured version Visualization version GIF version | ||
| Description: Provide a ring homomorphism between two univariate power series algebras over their respective base rings given a ring homomorphism between the two base rings. (Contributed by SN, 8-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| rhmpsr1.p | ⊢ 𝑃 = (PwSer1‘𝑅) | 
| rhmpsr1.q | ⊢ 𝑄 = (PwSer1‘𝑆) | 
| rhmpsr1.b | ⊢ 𝐵 = (Base‘𝑃) | 
| rhmpsr1.f | ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝)) | 
| rhmpsr1.h | ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆)) | 
| Ref | Expression | 
|---|---|
| rhmpsr1 | ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
| 2 | eqid 2737 | . . 3 ⊢ (1o mPwSer 𝑆) = (1o mPwSer 𝑆) | |
| 3 | rhmpsr1.p | . . . 4 ⊢ 𝑃 = (PwSer1‘𝑅) | |
| 4 | rhmpsr1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
| 5 | 3, 4, 1 | psr1bas2 22191 | . . 3 ⊢ 𝐵 = (Base‘(1o mPwSer 𝑅)) | 
| 6 | rhmpsr1.f | . . 3 ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝)) | |
| 7 | 1oex 8516 | . . . 4 ⊢ 1o ∈ V | |
| 8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → 1o ∈ V) | 
| 9 | rhmpsr1.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆)) | |
| 10 | 1, 2, 5, 6, 8, 9 | rhmpsr 42562 | . 2 ⊢ (𝜑 → 𝐹 ∈ ((1o mPwSer 𝑅) RingHom (1o mPwSer 𝑆))) | 
| 11 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
| 12 | 3, 11, 1 | psr1bas2 22191 | . . . 4 ⊢ (Base‘𝑃) = (Base‘(1o mPwSer 𝑅)) | 
| 13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → (Base‘𝑃) = (Base‘(1o mPwSer 𝑅))) | 
| 14 | rhmpsr1.q | . . . . 5 ⊢ 𝑄 = (PwSer1‘𝑆) | |
| 15 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
| 16 | 14, 15, 2 | psr1bas2 22191 | . . . 4 ⊢ (Base‘𝑄) = (Base‘(1o mPwSer 𝑆)) | 
| 17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → (Base‘𝑄) = (Base‘(1o mPwSer 𝑆))) | 
| 18 | eqidd 2738 | . . 3 ⊢ (𝜑 → (Base‘𝑃) = (Base‘𝑃)) | |
| 19 | eqidd 2738 | . . 3 ⊢ (𝜑 → (Base‘𝑄) = (Base‘𝑄)) | |
| 20 | eqid 2737 | . . . . . . 7 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
| 21 | 3, 1, 20 | psr1plusg 22222 | . . . . . 6 ⊢ (+g‘𝑃) = (+g‘(1o mPwSer 𝑅)) | 
| 22 | 21 | eqcomi 2746 | . . . . 5 ⊢ (+g‘(1o mPwSer 𝑅)) = (+g‘𝑃) | 
| 23 | 22 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (+g‘(1o mPwSer 𝑅)) = (+g‘𝑃)) | 
| 24 | 23 | oveqd 7448 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(+g‘(1o mPwSer 𝑅))𝑦) = (𝑥(+g‘𝑃)𝑦)) | 
| 25 | eqid 2737 | . . . . . . 7 ⊢ (+g‘𝑄) = (+g‘𝑄) | |
| 26 | 14, 2, 25 | psr1plusg 22222 | . . . . . 6 ⊢ (+g‘𝑄) = (+g‘(1o mPwSer 𝑆)) | 
| 27 | 26 | eqcomi 2746 | . . . . 5 ⊢ (+g‘(1o mPwSer 𝑆)) = (+g‘𝑄) | 
| 28 | 27 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑄) ∧ 𝑦 ∈ (Base‘𝑄))) → (+g‘(1o mPwSer 𝑆)) = (+g‘𝑄)) | 
| 29 | 28 | oveqd 7448 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑄) ∧ 𝑦 ∈ (Base‘𝑄))) → (𝑥(+g‘(1o mPwSer 𝑆))𝑦) = (𝑥(+g‘𝑄)𝑦)) | 
| 30 | eqid 2737 | . . . . . . 7 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
| 31 | 3, 1, 30 | psr1mulr 22224 | . . . . . 6 ⊢ (.r‘𝑃) = (.r‘(1o mPwSer 𝑅)) | 
| 32 | 31 | eqcomi 2746 | . . . . 5 ⊢ (.r‘(1o mPwSer 𝑅)) = (.r‘𝑃) | 
| 33 | 32 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (.r‘(1o mPwSer 𝑅)) = (.r‘𝑃)) | 
| 34 | 33 | oveqd 7448 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑃) ∧ 𝑦 ∈ (Base‘𝑃))) → (𝑥(.r‘(1o mPwSer 𝑅))𝑦) = (𝑥(.r‘𝑃)𝑦)) | 
| 35 | eqid 2737 | . . . . . . 7 ⊢ (.r‘𝑄) = (.r‘𝑄) | |
| 36 | 14, 2, 35 | psr1mulr 22224 | . . . . . 6 ⊢ (.r‘𝑄) = (.r‘(1o mPwSer 𝑆)) | 
| 37 | 36 | eqcomi 2746 | . . . . 5 ⊢ (.r‘(1o mPwSer 𝑆)) = (.r‘𝑄) | 
| 38 | 37 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑄) ∧ 𝑦 ∈ (Base‘𝑄))) → (.r‘(1o mPwSer 𝑆)) = (.r‘𝑄)) | 
| 39 | 38 | oveqd 7448 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑄) ∧ 𝑦 ∈ (Base‘𝑄))) → (𝑥(.r‘(1o mPwSer 𝑆))𝑦) = (𝑥(.r‘𝑄)𝑦)) | 
| 40 | 13, 17, 18, 19, 24, 29, 34, 39 | rhmpropd 20609 | . 2 ⊢ (𝜑 → ((1o mPwSer 𝑅) RingHom (1o mPwSer 𝑆)) = (𝑃 RingHom 𝑄)) | 
| 41 | 10, 40 | eleqtrd 2843 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ↦ cmpt 5225 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 1oc1o 8499 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 RingHom crh 20469 mPwSer cmps 21924 PwSer1cps1 22176 | 
| 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-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-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-rhm 20472 df-psr 21929 df-opsr 21933 df-psr1 22181 | 
| This theorem is referenced by: (None) | 
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