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Mirrors > Home > MPE Home > Th. List > nghmco | Structured version Visualization version GIF version |
Description: The composition of normed group homomorphisms is a normed group homomorphism. (Contributed by Mario Carneiro, 20-Oct-2015.) |
Ref | Expression |
---|---|
nghmco | ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 NGHom 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nghmrcl1 24569 | . . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp) | |
2 | 1 | adantl 481 | . 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑆 ∈ NrmGrp) |
3 | nghmrcl2 24570 | . . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝑈 ∈ NrmGrp) | |
4 | 3 | adantr 480 | . 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → 𝑈 ∈ NrmGrp) |
5 | nghmghm 24571 | . . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → 𝐹 ∈ (𝑇 GrpHom 𝑈)) | |
6 | nghmghm 24571 | . . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → 𝐺 ∈ (𝑆 GrpHom 𝑇)) | |
7 | ghmco 19157 | . . 3 ⊢ ((𝐹 ∈ (𝑇 GrpHom 𝑈) ∧ 𝐺 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈)) | |
8 | 5, 6, 7 | syl2an 595 | . 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈)) |
9 | eqid 2731 | . . . 4 ⊢ (𝑇 normOp 𝑈) = (𝑇 normOp 𝑈) | |
10 | 9 | nghmcl 24564 | . . 3 ⊢ (𝐹 ∈ (𝑇 NGHom 𝑈) → ((𝑇 normOp 𝑈)‘𝐹) ∈ ℝ) |
11 | eqid 2731 | . . . 4 ⊢ (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇) | |
12 | 11 | nghmcl 24564 | . . 3 ⊢ (𝐺 ∈ (𝑆 NGHom 𝑇) → ((𝑆 normOp 𝑇)‘𝐺) ∈ ℝ) |
13 | remulcl 11201 | . . 3 ⊢ ((((𝑇 normOp 𝑈)‘𝐹) ∈ ℝ ∧ ((𝑆 normOp 𝑇)‘𝐺) ∈ ℝ) → (((𝑇 normOp 𝑈)‘𝐹) · ((𝑆 normOp 𝑇)‘𝐺)) ∈ ℝ) | |
14 | 10, 12, 13 | syl2an 595 | . 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (((𝑇 normOp 𝑈)‘𝐹) · ((𝑆 normOp 𝑇)‘𝐺)) ∈ ℝ) |
15 | eqid 2731 | . . 3 ⊢ (𝑆 normOp 𝑈) = (𝑆 normOp 𝑈) | |
16 | 15, 9, 11 | nmoco 24574 | . 2 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → ((𝑆 normOp 𝑈)‘(𝐹 ∘ 𝐺)) ≤ (((𝑇 normOp 𝑈)‘𝐹) · ((𝑆 normOp 𝑇)‘𝐺))) |
17 | 15 | bddnghm 24563 | . 2 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑈 ∈ NrmGrp ∧ (𝐹 ∘ 𝐺) ∈ (𝑆 GrpHom 𝑈)) ∧ ((((𝑇 normOp 𝑈)‘𝐹) · ((𝑆 normOp 𝑇)‘𝐺)) ∈ ℝ ∧ ((𝑆 normOp 𝑈)‘(𝐹 ∘ 𝐺)) ≤ (((𝑇 normOp 𝑈)‘𝐹) · ((𝑆 normOp 𝑇)‘𝐺)))) → (𝐹 ∘ 𝐺) ∈ (𝑆 NGHom 𝑈)) |
18 | 2, 4, 8, 14, 16, 17 | syl32anc 1377 | 1 ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 NGHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 class class class wbr 5148 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7412 ℝcr 11115 · cmul 11121 ≤ cle 11256 GrpHom cghm 19134 NrmGrpcngp 24406 normOp cnmo 24542 NGHom cnghm 24543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ico 13337 df-0g 17394 df-topgen 17396 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-grp 18864 df-ghm 19135 df-psmet 21225 df-xmet 21226 df-met 21227 df-bl 21228 df-mopn 21229 df-top 22716 df-topon 22733 df-topsp 22755 df-bases 22769 df-xms 24146 df-ms 24147 df-nm 24411 df-ngp 24412 df-nmo 24545 df-nghm 24546 |
This theorem is referenced by: nmhmco 24593 |
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