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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 17926 | . . 3 ⊢ (𝜑 → (𝑌 + (𝑍 + 𝑊)) = (𝑍 + (𝑌 + 𝑊))) |
9 | 8 | oveq2d 7174 | . 2 ⊢ (𝜑 → (𝑋 + (𝑌 + (𝑍 + 𝑊))) = (𝑋 + (𝑍 + (𝑌 + 𝑊)))) |
10 | mnd4g.2 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | 1, 2 | mndcl 17921 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑍 + 𝑊) ∈ 𝐵) |
12 | 3, 5, 6, 11 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑍 + 𝑊) ∈ 𝐵) |
13 | 1, 2 | mndass 17922 | . . 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 1368 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊)))) |
15 | 1, 2 | mndcl 17921 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 + 𝑊) ∈ 𝐵) |
16 | 3, 4, 6, 15 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑌 + 𝑊) ∈ 𝐵) |
17 | 1, 2 | mndass 17922 | . . 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 1368 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊)))) |
19 | 9, 14, 18 | 3eqtr4d 2868 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 Mndcmnd 17913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-mgm 17854 df-sgrp 17903 df-mnd 17914 |
This theorem is referenced by: lsmsubm 18780 pj1ghm 18831 cmn4 18928 gsumzaddlem 19043 |
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