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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 7373 | . . 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 11362 | . . . 4 ⊢ (𝜑 → (𝐶 + 𝐷) = (𝐷 + 𝐶)) |
6 | 5 | oveq1d 7373 | . . 3 ⊢ (𝜑 → ((𝐶 + 𝐷) − (𝐷 + 𝐵)) = ((𝐷 + 𝐶) − (𝐷 + 𝐵))) |
7 | 2, 6 | eqtrd 2773 | . 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 11555 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐷 + 𝐵)) = (𝐴 − 𝐷)) |
11 | 4, 3, 9 | pnpcand 11554 | . 2 ⊢ (𝜑 → ((𝐷 + 𝐶) − (𝐷 + 𝐵)) = (𝐶 − 𝐵)) |
12 | 7, 10, 11 | 3eqtr3d 2781 | 1 ⊢ (𝜑 → (𝐴 − 𝐷) = (𝐶 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 (class class class)co 7358 ℂcc 11054 + caddc 11059 − cmin 11390 |
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-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-ltxr 11199 df-sub 11392 |
This theorem is referenced by: 2sqmod 26800 fmtnorec4 45827 |
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