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Mirrors > Home > ILE Home > Th. List > caovdilemd | GIF version |
Description: Lemma used by real number construction. (Contributed by Jim Kingdon, 16-Sep-2019.) |
Ref | Expression |
---|---|
caovdilemd.com | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) |
caovdilemd.distr | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧))) |
caovdilemd.ass | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))) |
caovdilemd.cl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆) |
caovdilemd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
caovdilemd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
caovdilemd.c | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
caovdilemd.d | ⊢ (𝜑 → 𝐷 ∈ 𝑆) |
caovdilemd.h | ⊢ (𝜑 → 𝐻 ∈ 𝑆) |
Ref | Expression |
---|---|
caovdilemd | ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovdilemd.distr | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧))) | |
2 | caovdilemd.cl | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆) | |
3 | caovdilemd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | caovdilemd.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
5 | 2, 3, 4 | caovcld 5654 | . . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) ∈ 𝑆) |
6 | caovdilemd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
7 | caovdilemd.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑆) | |
8 | 2, 6, 7 | caovcld 5654 | . . 3 ⊢ (𝜑 → (𝐵𝐺𝐷) ∈ 𝑆) |
9 | caovdilemd.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑆) | |
10 | 1, 5, 8, 9 | caovdird 5679 | . 2 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻))) |
11 | caovdilemd.ass | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))) | |
12 | 11, 3, 4, 9 | caovassd 5660 | . . 3 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻))) |
13 | 11, 6, 7, 9 | caovassd 5660 | . . 3 ⊢ (𝜑 → ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻))) |
14 | 12, 13 | oveq12d 5530 | . 2 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))) |
15 | 10, 14 | eqtrd 2072 | 1 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 (class class class)co 5512 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: caovlem2d 5693 addassnqg 6480 addassnq0 6560 axmulass 6947 |
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