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Mirrors > Home > ILE Home > Th. List > divalglemqt | GIF version |
Description: Lemma for divalg 11416. 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 5721 | . . 3 ⊢ (𝜑 → (𝑄 · 𝐷) = (𝑇 · 𝐷)) |
3 | divalglemqt.q | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℤ) | |
4 | divalglemqt.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℤ) | |
5 | 3, 4 | zmulcld 9031 | . . . 4 ⊢ (𝜑 → (𝑄 · 𝐷) ∈ ℤ) |
6 | 5 | zcnd 9026 | . . 3 ⊢ (𝜑 → (𝑄 · 𝐷) ∈ ℂ) |
7 | 2, 6 | eqeltrrd 2177 | . 2 ⊢ (𝜑 → (𝑇 · 𝐷) ∈ ℂ) |
8 | divalglemqt.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℤ) | |
9 | 8 | zcnd 9026 | . 2 ⊢ (𝜑 → 𝑅 ∈ ℂ) |
10 | divalglemqt.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℤ) | |
11 | 10 | zcnd 9026 | . 2 ⊢ (𝜑 → 𝑆 ∈ ℂ) |
12 | 2 | oveq1d 5721 | . . 3 ⊢ (𝜑 → ((𝑄 · 𝐷) + 𝑅) = ((𝑇 · 𝐷) + 𝑅)) |
13 | divalglemqt.eq | . . 3 ⊢ (𝜑 → ((𝑄 · 𝐷) + 𝑅) = ((𝑇 · 𝐷) + 𝑆)) | |
14 | 12, 13 | eqtr3d 2134 | . 2 ⊢ (𝜑 → ((𝑇 · 𝐷) + 𝑅) = ((𝑇 · 𝐷) + 𝑆)) |
15 | 7, 9, 11, 14 | addcanad 7819 | 1 ⊢ (𝜑 → 𝑅 = 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1299 ∈ wcel 1448 (class class class)co 5706 ℂcc 7498 + caddc 7503 · cmul 7505 ℤcz 8906 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-cnre 7606 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-sub 7806 df-neg 7807 df-inn 8579 df-n0 8830 df-z 8907 |
This theorem is referenced by: divalglemeunn 11413 divalglemeuneg 11415 |
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