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Mirrors > Home > ILE Home > Th. List > subaddeqd | 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 5903 | . . 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 8122 | . . . 4 ⊢ (𝜑 → (𝐶 + 𝐷) = (𝐷 + 𝐶)) |
6 | 5 | oveq1d 5903 | . . 3 ⊢ (𝜑 → ((𝐶 + 𝐷) − (𝐷 + 𝐵)) = ((𝐷 + 𝐶) − (𝐷 + 𝐵))) |
7 | 2, 6 | eqtrd 2220 | . 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 8320 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐷 + 𝐵)) = (𝐴 − 𝐷)) |
11 | 4, 3, 9 | pnpcand 8319 | . 2 ⊢ (𝜑 → ((𝐷 + 𝐶) − (𝐷 + 𝐵)) = (𝐶 − 𝐵)) |
12 | 7, 10, 11 | 3eqtr3d 2228 | 1 ⊢ (𝜑 → (𝐴 − 𝐷) = (𝐶 − 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 (class class class)co 5888 ℂcc 7823 + caddc 7828 − cmin 8142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-setind 4548 ax-resscn 7917 ax-1cn 7918 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-distr 7929 ax-i2m1 7930 ax-0id 7933 ax-rnegex 7934 ax-cnre 7936 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-sub 8144 |
This theorem is referenced by: (None) |
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