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Mirrors > Home > MPE Home > Th. List > Mathboxes > mhmcoaddmpl | Structured version Visualization version GIF version |
Description: Show that the ring homomorphism in rhmmpl 41698 preserves addition. (Contributed by SN, 8-Feb-2025.) |
Ref | Expression |
---|---|
mhmcoaddmpl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mhmcoaddmpl.q | ⊢ 𝑄 = (𝐼 mPoly 𝑆) |
mhmcoaddmpl.b | ⊢ 𝐵 = (Base‘𝑃) |
mhmcoaddmpl.c | ⊢ 𝐶 = (Base‘𝑄) |
mhmcoaddmpl.1 | ⊢ + = (+g‘𝑃) |
mhmcoaddmpl.2 | ⊢ ✚ = (+g‘𝑄) |
mhmcoaddmpl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mhmcoaddmpl.h | ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆)) |
mhmcoaddmpl.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
mhmcoaddmpl.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
Ref | Expression |
---|---|
mhmcoaddmpl | ⊢ (𝜑 → (𝐻 ∘ (𝐹 + 𝐺)) = ((𝐻 ∘ 𝐹) ✚ (𝐻 ∘ 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhmcoaddmpl.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆)) | |
2 | fvexd 6906 | . . . 4 ⊢ (𝜑 → (Base‘𝑅) ∈ V) | |
3 | ovex 7447 | . . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
4 | 3 | rabex 5328 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) |
6 | mhmcoaddmpl.p | . . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
7 | eqid 2727 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | mhmcoaddmpl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
9 | eqid 2727 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
10 | mhmcoaddmpl.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
11 | 6, 7, 8, 9, 10 | mplelf 21918 | . . . 4 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
12 | 2, 5, 11 | elmapdd 8849 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
13 | mhmcoaddmpl.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
14 | 6, 7, 8, 9, 13 | mplelf 21918 | . . . 4 ⊢ (𝜑 → 𝐺:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
15 | 2, 5, 14 | elmapdd 8849 | . . 3 ⊢ (𝜑 → 𝐺 ∈ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
16 | eqid 2727 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
17 | eqid 2727 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
18 | 7, 16, 17 | mhmvlin 22273 | . . 3 ⊢ ((𝐻 ∈ (𝑅 MndHom 𝑆) ∧ 𝐹 ∈ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝐺 ∈ ((Base‘𝑅) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) → (𝐻 ∘ (𝐹 ∘f (+g‘𝑅)𝐺)) = ((𝐻 ∘ 𝐹) ∘f (+g‘𝑆)(𝐻 ∘ 𝐺))) |
19 | 1, 12, 15, 18 | syl3anc 1369 | . 2 ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f (+g‘𝑅)𝐺)) = ((𝐻 ∘ 𝐹) ∘f (+g‘𝑆)(𝐻 ∘ 𝐺))) |
20 | mhmcoaddmpl.1 | . . . 4 ⊢ + = (+g‘𝑃) | |
21 | 6, 8, 16, 20, 10, 13 | mpladd 21929 | . . 3 ⊢ (𝜑 → (𝐹 + 𝐺) = (𝐹 ∘f (+g‘𝑅)𝐺)) |
22 | 21 | coeq2d 5859 | . 2 ⊢ (𝜑 → (𝐻 ∘ (𝐹 + 𝐺)) = (𝐻 ∘ (𝐹 ∘f (+g‘𝑅)𝐺))) |
23 | mhmcoaddmpl.q | . . 3 ⊢ 𝑄 = (𝐼 mPoly 𝑆) | |
24 | mhmcoaddmpl.c | . . 3 ⊢ 𝐶 = (Base‘𝑄) | |
25 | mhmcoaddmpl.2 | . . 3 ⊢ ✚ = (+g‘𝑄) | |
26 | mhmcoaddmpl.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
27 | 6, 23, 8, 24, 26, 1, 10 | mhmcompl 41693 | . . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶) |
28 | 6, 23, 8, 24, 26, 1, 13 | mhmcompl 41693 | . . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐺) ∈ 𝐶) |
29 | 23, 24, 17, 25, 27, 28 | mpladd 21929 | . 2 ⊢ (𝜑 → ((𝐻 ∘ 𝐹) ✚ (𝐻 ∘ 𝐺)) = ((𝐻 ∘ 𝐹) ∘f (+g‘𝑆)(𝐻 ∘ 𝐺))) |
30 | 19, 22, 29 | 3eqtr4d 2777 | 1 ⊢ (𝜑 → (𝐻 ∘ (𝐹 + 𝐺)) = ((𝐻 ∘ 𝐹) ✚ (𝐻 ∘ 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {crab 3427 Vcvv 3469 ◡ccnv 5671 “ cima 5675 ∘ ccom 5676 ‘cfv 6542 (class class class)co 7414 ∘f cof 7675 ↑m cmap 8834 Fincfn 8953 ℕcn 12228 ℕ0cn0 12488 Basecbs 17165 +gcplusg 17218 MndHom cmhm 18723 mPoly cmpl 21819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-tset 17237 df-0g 17408 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-psr 21822 df-mpl 21824 |
This theorem is referenced by: rhmmpl 41698 selvadd 41733 |
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