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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 5932 | . . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) ∈ 𝑆) |
6 | caovdilemd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
7 | caovdilemd.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑆) | |
8 | 2, 6, 7 | caovcld 5932 | . . 3 ⊢ (𝜑 → (𝐵𝐺𝐷) ∈ 𝑆) |
9 | caovdilemd.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑆) | |
10 | 1, 5, 8, 9 | caovdird 5957 | . 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 5938 | . . 3 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻))) |
13 | 11, 6, 7, 9 | caovassd 5938 | . . 3 ⊢ (𝜑 → ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻))) |
14 | 12, 13 | oveq12d 5800 | . 2 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))) |
15 | 10, 14 | eqtrd 2173 | 1 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 (class class class)co 5782 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 |
This theorem is referenced by: caovlem2d 5971 addassnqg 7214 addassnq0 7294 axmulass 7705 |
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