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Mirrors > Home > MPE Home > Th. List > Mathboxes > mhmcopsr | Structured version Visualization version GIF version |
Description: The composition of a monoid homomorphism and a power series is a power series. (Contributed by SN, 18-May-2025.) |
Ref | Expression |
---|---|
mhmcopsr.p | ⊢ 𝑃 = (𝐼 mPwSer 𝑅) |
mhmcopsr.q | ⊢ 𝑄 = (𝐼 mPwSer 𝑆) |
mhmcopsr.b | ⊢ 𝐵 = (Base‘𝑃) |
mhmcopsr.c | ⊢ 𝐶 = (Base‘𝑄) |
mhmcopsr.h | ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆)) |
mhmcopsr.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
Ref | Expression |
---|---|
mhmcopsr | ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6922 | . . 3 ⊢ (𝜑 → (Base‘𝑆) ∈ V) | |
2 | ovex 7464 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
3 | 2 | rabex 5345 | . . . 4 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V) |
5 | mhmcopsr.h | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆)) | |
6 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
8 | 6, 7 | mhmf 18815 | . . . . 5 ⊢ (𝐻 ∈ (𝑅 MndHom 𝑆) → 𝐻:(Base‘𝑅)⟶(Base‘𝑆)) |
9 | 5, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐻:(Base‘𝑅)⟶(Base‘𝑆)) |
10 | mhmcopsr.p | . . . . 5 ⊢ 𝑃 = (𝐼 mPwSer 𝑅) | |
11 | eqid 2735 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
12 | mhmcopsr.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
13 | mhmcopsr.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
14 | 10, 6, 11, 12, 13 | psrelbas 21972 | . . . 4 ⊢ (𝜑 → 𝐹:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
15 | 9, 14 | fcod 6762 | . . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑆)) |
16 | 1, 4, 15 | elmapdd 8880 | . 2 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Base‘𝑆) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
17 | mhmcopsr.q | . . 3 ⊢ 𝑄 = (𝐼 mPwSer 𝑆) | |
18 | mhmcopsr.c | . . 3 ⊢ 𝐶 = (Base‘𝑄) | |
19 | reldmpsr 21952 | . . . . . 6 ⊢ Rel dom mPwSer | |
20 | 19, 10, 12 | elbasov 17252 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
21 | 13, 20 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
22 | 21 | simpld 494 | . . 3 ⊢ (𝜑 → 𝐼 ∈ V) |
23 | 17, 7, 11, 18, 22 | psrbas 21971 | . 2 ⊢ (𝜑 → 𝐶 = ((Base‘𝑆) ↑m {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})) |
24 | 16, 23 | eleqtrrd 2842 | 1 ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ◡ccnv 5688 “ cima 5692 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Fincfn 8984 ℕcn 12264 ℕ0cn0 12524 Basecbs 17245 MndHom cmhm 18807 mPwSer cmps 21942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-tset 17317 df-mhm 18809 df-psr 21947 |
This theorem is referenced by: mhmcoaddpsr 42537 rhmcomulpsr 42538 rhmpsr 42539 |
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