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Mirrors > Home > MPE Home > Th. List > Mathboxes > subadd4b | Structured version Visualization version GIF version |
Description: Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
subadd4b.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
subadd4b.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
subadd4b.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
subadd4b.4 | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
Ref | Expression |
---|---|
subadd4b | ⊢ (𝜑 → ((𝐴 − 𝐵) + (𝐶 − 𝐷)) = ((𝐴 − 𝐷) + (𝐶 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subadd4b.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | subadd4b.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | subadd4b.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
4 | subadd4b.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
5 | 1, 2, 3, 4 | subadd4d 11047 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) − (𝐷 − 𝐶)) = ((𝐴 + 𝐶) − (𝐵 + 𝐷))) |
6 | 1, 2 | subcld 10999 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
7 | 6, 3, 4 | subsub2d 11028 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) − (𝐷 − 𝐶)) = ((𝐴 − 𝐵) + (𝐶 − 𝐷))) |
8 | 2, 3 | addcomd 10844 | . . . 4 ⊢ (𝜑 → (𝐵 + 𝐷) = (𝐷 + 𝐵)) |
9 | 8 | oveq2d 7174 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐶) − (𝐵 + 𝐷)) = ((𝐴 + 𝐶) − (𝐷 + 𝐵))) |
10 | 1, 4, 3, 2 | addsub4d 11046 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐶) − (𝐷 + 𝐵)) = ((𝐴 − 𝐷) + (𝐶 − 𝐵))) |
11 | 9, 10 | eqtrd 2858 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) − (𝐵 + 𝐷)) = ((𝐴 − 𝐷) + (𝐶 − 𝐵))) |
12 | 5, 7, 11 | 3eqtr3d 2866 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) + (𝐶 − 𝐷)) = ((𝐴 − 𝐷) + (𝐶 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 + caddc 10542 − cmin 10872 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 |
This theorem is referenced by: fourierdlem42 42441 fourierdlem107 42505 |
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