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Mirrors > Home > HSE Home > Th. List > normlem8 | Structured version Visualization version GIF version |
Description: Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
normlem8.1 | ⊢ 𝐴 ∈ ℋ |
normlem8.2 | ⊢ 𝐵 ∈ ℋ |
normlem8.3 | ⊢ 𝐶 ∈ ℋ |
normlem8.4 | ⊢ 𝐷 ∈ ℋ |
Ref | Expression |
---|---|
normlem8 | ⊢ ((𝐴 +ℎ 𝐵) ·ih (𝐶 +ℎ 𝐷)) = (((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐷)) + ((𝐴 ·ih 𝐷) + (𝐵 ·ih 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | normlem8.1 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
2 | normlem8.3 | . . . 4 ⊢ 𝐶 ∈ ℋ | |
3 | normlem8.4 | . . . 4 ⊢ 𝐷 ∈ ℋ | |
4 | his7 28472 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐴 ·ih (𝐶 +ℎ 𝐷)) = ((𝐴 ·ih 𝐶) + (𝐴 ·ih 𝐷))) | |
5 | 1, 2, 3, 4 | mp3an 1586 | . . 3 ⊢ (𝐴 ·ih (𝐶 +ℎ 𝐷)) = ((𝐴 ·ih 𝐶) + (𝐴 ·ih 𝐷)) |
6 | normlem8.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
7 | his7 28472 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐵 ·ih (𝐶 +ℎ 𝐷)) = ((𝐵 ·ih 𝐶) + (𝐵 ·ih 𝐷))) | |
8 | 6, 2, 3, 7 | mp3an 1586 | . . 3 ⊢ (𝐵 ·ih (𝐶 +ℎ 𝐷)) = ((𝐵 ·ih 𝐶) + (𝐵 ·ih 𝐷)) |
9 | 5, 8 | oveq12i 6890 | . 2 ⊢ ((𝐴 ·ih (𝐶 +ℎ 𝐷)) + (𝐵 ·ih (𝐶 +ℎ 𝐷))) = (((𝐴 ·ih 𝐶) + (𝐴 ·ih 𝐷)) + ((𝐵 ·ih 𝐶) + (𝐵 ·ih 𝐷))) |
10 | 2, 3 | hvaddcli 28400 | . . 3 ⊢ (𝐶 +ℎ 𝐷) ∈ ℋ |
11 | ax-his2 28465 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (𝐶 +ℎ 𝐷) ∈ ℋ) → ((𝐴 +ℎ 𝐵) ·ih (𝐶 +ℎ 𝐷)) = ((𝐴 ·ih (𝐶 +ℎ 𝐷)) + (𝐵 ·ih (𝐶 +ℎ 𝐷)))) | |
12 | 1, 6, 10, 11 | mp3an 1586 | . 2 ⊢ ((𝐴 +ℎ 𝐵) ·ih (𝐶 +ℎ 𝐷)) = ((𝐴 ·ih (𝐶 +ℎ 𝐷)) + (𝐵 ·ih (𝐶 +ℎ 𝐷))) |
13 | 1, 2 | hicli 28463 | . . 3 ⊢ (𝐴 ·ih 𝐶) ∈ ℂ |
14 | 6, 3 | hicli 28463 | . . 3 ⊢ (𝐵 ·ih 𝐷) ∈ ℂ |
15 | 1, 3 | hicli 28463 | . . 3 ⊢ (𝐴 ·ih 𝐷) ∈ ℂ |
16 | 6, 2 | hicli 28463 | . . 3 ⊢ (𝐵 ·ih 𝐶) ∈ ℂ |
17 | 13, 14, 15, 16 | add42i 10551 | . 2 ⊢ (((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐷)) + ((𝐴 ·ih 𝐷) + (𝐵 ·ih 𝐶))) = (((𝐴 ·ih 𝐶) + (𝐴 ·ih 𝐷)) + ((𝐵 ·ih 𝐶) + (𝐵 ·ih 𝐷))) |
18 | 9, 12, 17 | 3eqtr4i 2831 | 1 ⊢ ((𝐴 +ℎ 𝐵) ·ih (𝐶 +ℎ 𝐷)) = (((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐷)) + ((𝐴 ·ih 𝐷) + (𝐵 ·ih 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 (class class class)co 6878 + caddc 10227 ℋchba 28301 +ℎ cva 28302 ·ih csp 28304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-hfvadd 28382 ax-hfi 28461 ax-his1 28464 ax-his2 28465 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-2 11376 df-cj 14180 df-re 14181 df-im 14182 |
This theorem is referenced by: normlem9 28500 norm-ii-i 28519 normpythi 28524 normpari 28536 polid2i 28539 |
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