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Mirrors > Home > MPE Home > Th. List > pwsdiagghm | Structured version Visualization version GIF version |
Description: Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
pwsdiagghm.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsdiagghm.b | ⊢ 𝐵 = (Base‘𝑅) |
pwsdiagghm.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) |
Ref | Expression |
---|---|
pwsdiagghm | ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 GrpHom 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 18821 | . . 3 ⊢ (𝑅 ∈ Grp → 𝑅 ∈ Mnd) | |
2 | pwsdiagghm.y | . . . 4 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
3 | pwsdiagghm.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | pwsdiagghm.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) | |
5 | 2, 3, 4 | pwsdiagmhm 18707 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 MndHom 𝑌)) |
6 | 1, 5 | sylan 581 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 MndHom 𝑌)) |
7 | 2 | pwsgrp 18930 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑊) → 𝑌 ∈ Grp) |
8 | ghmmhmb 19096 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝑌 ∈ Grp) → (𝑅 GrpHom 𝑌) = (𝑅 MndHom 𝑌)) | |
9 | 7, 8 | syldan 592 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑊) → (𝑅 GrpHom 𝑌) = (𝑅 MndHom 𝑌)) |
10 | 6, 9 | eleqtrrd 2837 | 1 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 GrpHom 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4626 ↦ cmpt 5229 × cxp 5672 ‘cfv 6539 (class class class)co 7403 Basecbs 17139 ↑s cpws 17387 Mndcmnd 18620 MndHom cmhm 18664 Grpcgrp 18814 GrpHom cghm 19082 |
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 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-map 8817 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-z 12554 df-dec 12673 df-uz 12818 df-fz 13480 df-struct 17075 df-slot 17110 df-ndx 17122 df-base 17140 df-plusg 17205 df-mulr 17206 df-sca 17208 df-vsca 17209 df-ip 17210 df-tset 17211 df-ple 17212 df-ds 17214 df-hom 17216 df-cco 17217 df-0g 17382 df-prds 17388 df-pws 17390 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-mhm 18666 df-grp 18817 df-minusg 18818 df-ghm 19083 |
This theorem is referenced by: pwsdiagrhm 20386 pwsdiaglmhm 20655 |
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