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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 29918 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐴 ·ih (𝐶 +ℎ 𝐷)) = ((𝐴 ·ih 𝐶) + (𝐴 ·ih 𝐷))) | |
5 | 1, 2, 3, 4 | mp3an 1461 | . . 3 ⊢ (𝐴 ·ih (𝐶 +ℎ 𝐷)) = ((𝐴 ·ih 𝐶) + (𝐴 ·ih 𝐷)) |
6 | normlem8.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
7 | his7 29918 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐵 ·ih (𝐶 +ℎ 𝐷)) = ((𝐵 ·ih 𝐶) + (𝐵 ·ih 𝐷))) | |
8 | 6, 2, 3, 7 | mp3an 1461 | . . 3 ⊢ (𝐵 ·ih (𝐶 +ℎ 𝐷)) = ((𝐵 ·ih 𝐶) + (𝐵 ·ih 𝐷)) |
9 | 5, 8 | oveq12i 7365 | . 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 29846 | . . 3 ⊢ (𝐶 +ℎ 𝐷) ∈ ℋ |
11 | ax-his2 29911 | . . 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 1461 | . 2 ⊢ ((𝐴 +ℎ 𝐵) ·ih (𝐶 +ℎ 𝐷)) = ((𝐴 ·ih (𝐶 +ℎ 𝐷)) + (𝐵 ·ih (𝐶 +ℎ 𝐷))) |
13 | 1, 2 | hicli 29909 | . . 3 ⊢ (𝐴 ·ih 𝐶) ∈ ℂ |
14 | 6, 3 | hicli 29909 | . . 3 ⊢ (𝐵 ·ih 𝐷) ∈ ℂ |
15 | 1, 3 | hicli 29909 | . . 3 ⊢ (𝐴 ·ih 𝐷) ∈ ℂ |
16 | 6, 2 | hicli 29909 | . . 3 ⊢ (𝐵 ·ih 𝐶) ∈ ℂ |
17 | 13, 14, 15, 16 | add42i 11376 | . 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 2774 | 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 1541 ∈ wcel 2106 (class class class)co 7353 + caddc 11050 ℋchba 29747 +ℎ cva 29748 ·ih csp 29750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-hfvadd 29828 ax-hfi 29907 ax-his1 29910 ax-his2 29911 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-2 12212 df-cj 14976 df-re 14977 df-im 14978 |
This theorem is referenced by: normlem9 29946 norm-ii-i 29965 normpythi 29970 normpari 29982 polid2i 29985 |
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