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Mirrors > Home > MPE Home > Th. List > subaddeqd | Structured version Visualization version GIF version |
Description: Transfer two terms of a subtraction to an addition in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.) |
Ref | Expression |
---|---|
subaddeqd.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
subaddeqd.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
subaddeqd.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
subaddeqd.d | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
subaddeqd.1 | ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷)) |
Ref | Expression |
---|---|
subaddeqd | ⊢ (𝜑 → (𝐴 − 𝐷) = (𝐶 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subaddeqd.1 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷)) | |
2 | 1 | oveq1d 7429 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐷 + 𝐵)) = ((𝐶 + 𝐷) − (𝐷 + 𝐵))) |
3 | subaddeqd.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | subaddeqd.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
5 | 3, 4 | addcomd 11432 | . . . 4 ⊢ (𝜑 → (𝐶 + 𝐷) = (𝐷 + 𝐶)) |
6 | 5 | oveq1d 7429 | . . 3 ⊢ (𝜑 → ((𝐶 + 𝐷) − (𝐷 + 𝐵)) = ((𝐷 + 𝐶) − (𝐷 + 𝐵))) |
7 | 2, 6 | eqtrd 2767 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐷 + 𝐵)) = ((𝐷 + 𝐶) − (𝐷 + 𝐵))) |
8 | subaddeqd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
9 | subaddeqd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
10 | 8, 4, 9 | pnpcan2d 11625 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐷 + 𝐵)) = (𝐴 − 𝐷)) |
11 | 4, 3, 9 | pnpcand 11624 | . 2 ⊢ (𝜑 → ((𝐷 + 𝐶) − (𝐷 + 𝐵)) = (𝐶 − 𝐵)) |
12 | 7, 10, 11 | 3eqtr3d 2775 | 1 ⊢ (𝜑 → (𝐴 − 𝐷) = (𝐶 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 (class class class)co 7414 ℂcc 11122 + caddc 11127 − cmin 11460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-ltxr 11269 df-sub 11462 |
This theorem is referenced by: 2sqmod 27343 fmtnorec4 46802 |
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