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Mirrors > Home > ILE Home > Th. List > divalglemqt | GIF version |
Description: Lemma for divalg 11610. The 𝑄 = 𝑇 case involved in showing uniqueness. (Contributed by Jim Kingdon, 5-Dec-2021.) |
Ref | Expression |
---|---|
divalglemqt.d | ⊢ (𝜑 → 𝐷 ∈ ℤ) |
divalglemqt.r | ⊢ (𝜑 → 𝑅 ∈ ℤ) |
divalglemqt.s | ⊢ (𝜑 → 𝑆 ∈ ℤ) |
divalglemqt.q | ⊢ (𝜑 → 𝑄 ∈ ℤ) |
divalglemqt.t | ⊢ (𝜑 → 𝑇 ∈ ℤ) |
divalglemqt.qt | ⊢ (𝜑 → 𝑄 = 𝑇) |
divalglemqt.eq | ⊢ (𝜑 → ((𝑄 · 𝐷) + 𝑅) = ((𝑇 · 𝐷) + 𝑆)) |
Ref | Expression |
---|---|
divalglemqt | ⊢ (𝜑 → 𝑅 = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divalglemqt.qt | . . . 4 ⊢ (𝜑 → 𝑄 = 𝑇) | |
2 | 1 | oveq1d 5782 | . . 3 ⊢ (𝜑 → (𝑄 · 𝐷) = (𝑇 · 𝐷)) |
3 | divalglemqt.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℤ) | |
4 | divalglemqt.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℤ) | |
5 | 3, 4 | zmulcld 9172 | . . . 4 ⊢ (𝜑 → (𝑄 · 𝐷) ∈ ℤ) |
6 | 5 | zcnd 9167 | . . 3 ⊢ (𝜑 → (𝑄 · 𝐷) ∈ ℂ) |
7 | 2, 6 | eqeltrrd 2215 | . 2 ⊢ (𝜑 → (𝑇 · 𝐷) ∈ ℂ) |
8 | divalglemqt.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℤ) | |
9 | 8 | zcnd 9167 | . 2 ⊢ (𝜑 → 𝑅 ∈ ℂ) |
10 | divalglemqt.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℤ) | |
11 | 10 | zcnd 9167 | . 2 ⊢ (𝜑 → 𝑆 ∈ ℂ) |
12 | 2 | oveq1d 5782 | . . 3 ⊢ (𝜑 → ((𝑄 · 𝐷) + 𝑅) = ((𝑇 · 𝐷) + 𝑅)) |
13 | divalglemqt.eq | . . 3 ⊢ (𝜑 → ((𝑄 · 𝐷) + 𝑅) = ((𝑇 · 𝐷) + 𝑆)) | |
14 | 12, 13 | eqtr3d 2172 | . 2 ⊢ (𝜑 → ((𝑇 · 𝐷) + 𝑅) = ((𝑇 · 𝐷) + 𝑆)) |
15 | 7, 9, 11, 14 | addcanad 7941 | 1 ⊢ (𝜑 → 𝑅 = 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 (class class class)co 5767 ℂcc 7611 + caddc 7616 · cmul 7618 ℤcz 9047 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 |
This theorem is referenced by: divalglemeunn 11607 divalglemeuneg 11609 |
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