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Mirrors > Home > MPE Home > Th. List > add4d | Structured version Visualization version GIF version |
Description: Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
addd.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
add4d.4 | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
Ref | Expression |
---|---|
add4d | ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | addd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | addd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | add4d.4 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
5 | add4 10849 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))) | |
6 | 1, 2, 3, 4, 5 | syl22anc 837 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 + caddc 10529 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 |
This theorem is referenced by: binom3 13581 readd 14477 imadd 14485 bhmafibid1cn 14815 bhmafibid2cn 14816 bpoly4 15405 efi4p 15482 sadcaddlem 15796 sadadd2lem 15798 4sqlem11 16281 vdwlem6 16312 cphipval2 23845 ovolunlem1a 24100 uniioombllem3 24189 itgadd 24428 itgmulc2 24437 tanarg 25210 binom4 25436 dquart 25439 quart1 25442 axeuclidlem 26756 axcontlem8 26765 golem1 30054 archiabllem2c 30874 madjusmdetlem4 31183 itgaddnclem2 35116 itgaddnc 35117 itgmulc2nc 35125 dvnmul 42585 stoweidlem13 42655 sge0xaddlem1 43072 opoeALTV 44201 itsclc0xyqsolr 45183 |
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