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Mirrors > Home > MPE Home > Th. List > mnd4g | Structured version Visualization version GIF version |
Description: Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
mndcl.b | ⊢ 𝐵 = (Base‘𝐺) |
mndcl.p | ⊢ + = (+g‘𝐺) |
mnd4g.1 | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
mnd4g.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
mnd4g.3 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
mnd4g.4 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
mnd4g.5 | ⊢ (𝜑 → 𝑊 ∈ 𝐵) |
mnd4g.6 | ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌)) |
Ref | Expression |
---|---|
mnd4g | ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | mndcl.p | . . . 4 ⊢ + = (+g‘𝐺) | |
3 | mnd4g.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
4 | mnd4g.3 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
5 | mnd4g.4 | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
6 | mnd4g.5 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵) | |
7 | mnd4g.6 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | mnd12g 18033 | . . 3 ⊢ (𝜑 → (𝑌 + (𝑍 + 𝑊)) = (𝑍 + (𝑌 + 𝑊))) |
9 | 8 | oveq2d 7180 | . 2 ⊢ (𝜑 → (𝑋 + (𝑌 + (𝑍 + 𝑊))) = (𝑋 + (𝑍 + (𝑌 + 𝑊)))) |
10 | mnd4g.2 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 1, 2 | mndcl 18028 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑍 + 𝑊) ∈ 𝐵) |
12 | 3, 5, 6, 11 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (𝑍 + 𝑊) ∈ 𝐵) |
13 | 1, 2 | mndass 18029 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊)))) |
14 | 3, 10, 4, 12, 13 | syl13anc 1373 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊)))) |
15 | 1, 2 | mndcl 18028 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 + 𝑊) ∈ 𝐵) |
16 | 3, 4, 6, 15 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (𝑌 + 𝑊) ∈ 𝐵) |
17 | 1, 2 | mndass 18029 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ (𝑌 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊)))) |
18 | 3, 10, 5, 16, 17 | syl13anc 1373 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊)))) |
19 | 9, 14, 18 | 3eqtr4d 2783 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 +gcplusg 16661 Mndcmnd 18020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-nul 5171 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-iota 6291 df-fv 6341 df-ov 7167 df-mgm 17961 df-sgrp 18010 df-mnd 18021 |
This theorem is referenced by: lsmsubm 18889 pj1ghm 18940 cmn4 19037 gsumzaddlem 19153 |
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