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Mirrors > Home > MPE Home > Th. List > nvadd4 | Structured version Visualization version GIF version |
Description: Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvgcl.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvgcl.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
Ref | Expression |
---|---|
nvadd4 | ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvgcl.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
2 | 1 | nvablo 29846 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp) |
3 | nvgcl.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 3, 1 | bafval 29834 | . . 3 ⊢ 𝑋 = ran 𝐺 |
5 | 4 | ablo4 29780 | . 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷))) |
6 | 2, 5 | syl3an1 1164 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6539 (class class class)co 7403 AbelOpcablo 29774 NrmCVeccnv 29814 +𝑣 cpv 29815 BaseSetcba 29816 |
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-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3063 df-rex 3072 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-nul 4321 df-if 4527 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 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-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7406 df-oprab 7407 df-1st 7969 df-2nd 7970 df-grpo 29723 df-ablo 29775 df-vc 29789 df-nv 29822 df-va 29825 df-ba 29826 df-sm 29827 df-0v 29828 df-nmcv 29830 |
This theorem is referenced by: nvaddsub4 29887 |
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