| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > prdscmnd | Structured version Visualization version GIF version | ||
| Description: The product of a family of commutative monoids is commutative. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
| Ref | Expression |
|---|---|
| prdscmnd.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
| prdscmnd.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
| prdscmnd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
| prdscmnd.r | ⊢ (𝜑 → 𝑅:𝐼⟶CMnd) |
| Ref | Expression |
|---|---|
| prdscmnd | ⊢ (𝜑 → 𝑌 ∈ CMnd) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqidd 2738 | . 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌)) | |
| 2 | eqidd 2738 | . 2 ⊢ (𝜑 → (+g‘𝑌) = (+g‘𝑌)) | |
| 3 | prdscmnd.y | . . 3 ⊢ 𝑌 = (𝑆Xs𝑅) | |
| 4 | prdscmnd.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
| 5 | prdscmnd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
| 6 | prdscmnd.r | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶CMnd) | |
| 7 | cmnmnd 19815 | . . . . 5 ⊢ (𝑎 ∈ CMnd → 𝑎 ∈ Mnd) | |
| 8 | 7 | ssriv 3987 | . . . 4 ⊢ CMnd ⊆ Mnd |
| 9 | fss 6752 | . . . 4 ⊢ ((𝑅:𝐼⟶CMnd ∧ CMnd ⊆ Mnd) → 𝑅:𝐼⟶Mnd) | |
| 10 | 6, 8, 9 | sylancl 586 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) |
| 11 | 3, 4, 5, 10 | prdsmndd 18783 | . 2 ⊢ (𝜑 → 𝑌 ∈ Mnd) |
| 12 | 6 | 3ad2ant1 1134 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶CMnd) |
| 13 | 12 | ffvelcdmda 7104 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) ∧ 𝑐 ∈ 𝐼) → (𝑅‘𝑐) ∈ CMnd) |
| 14 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
| 15 | 5 | elexd 3504 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ V) |
| 16 | 15 | 3ad2ant1 1134 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → 𝑆 ∈ V) |
| 17 | 16 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) ∧ 𝑐 ∈ 𝐼) → 𝑆 ∈ V) |
| 18 | 4 | elexd 3504 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V) |
| 19 | 18 | 3ad2ant1 1134 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → 𝐼 ∈ V) |
| 20 | 19 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) ∧ 𝑐 ∈ 𝐼) → 𝐼 ∈ V) |
| 21 | 6 | ffnd 6737 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼) |
| 22 | 21 | 3ad2ant1 1134 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼) |
| 23 | 22 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) ∧ 𝑐 ∈ 𝐼) → 𝑅 Fn 𝐼) |
| 24 | simpl2 1193 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) ∧ 𝑐 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑌)) | |
| 25 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) ∧ 𝑐 ∈ 𝐼) → 𝑐 ∈ 𝐼) | |
| 26 | 3, 14, 17, 20, 23, 24, 25 | prdsbasprj 17517 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) ∧ 𝑐 ∈ 𝐼) → (𝑎‘𝑐) ∈ (Base‘(𝑅‘𝑐))) |
| 27 | simpl3 1194 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) ∧ 𝑐 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑌)) | |
| 28 | 3, 14, 17, 20, 23, 27, 25 | prdsbasprj 17517 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) ∧ 𝑐 ∈ 𝐼) → (𝑏‘𝑐) ∈ (Base‘(𝑅‘𝑐))) |
| 29 | eqid 2737 | . . . . . 6 ⊢ (Base‘(𝑅‘𝑐)) = (Base‘(𝑅‘𝑐)) | |
| 30 | eqid 2737 | . . . . . 6 ⊢ (+g‘(𝑅‘𝑐)) = (+g‘(𝑅‘𝑐)) | |
| 31 | 29, 30 | cmncom 19816 | . . . . 5 ⊢ (((𝑅‘𝑐) ∈ CMnd ∧ (𝑎‘𝑐) ∈ (Base‘(𝑅‘𝑐)) ∧ (𝑏‘𝑐) ∈ (Base‘(𝑅‘𝑐))) → ((𝑎‘𝑐)(+g‘(𝑅‘𝑐))(𝑏‘𝑐)) = ((𝑏‘𝑐)(+g‘(𝑅‘𝑐))(𝑎‘𝑐))) |
| 32 | 13, 26, 28, 31 | syl3anc 1373 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) ∧ 𝑐 ∈ 𝐼) → ((𝑎‘𝑐)(+g‘(𝑅‘𝑐))(𝑏‘𝑐)) = ((𝑏‘𝑐)(+g‘(𝑅‘𝑐))(𝑎‘𝑐))) |
| 33 | 32 | mpteq2dva 5242 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑐 ∈ 𝐼 ↦ ((𝑎‘𝑐)(+g‘(𝑅‘𝑐))(𝑏‘𝑐))) = (𝑐 ∈ 𝐼 ↦ ((𝑏‘𝑐)(+g‘(𝑅‘𝑐))(𝑎‘𝑐)))) |
| 34 | simp2 1138 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌)) | |
| 35 | simp3 1139 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → 𝑏 ∈ (Base‘𝑌)) | |
| 36 | eqid 2737 | . . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
| 37 | 3, 14, 16, 19, 22, 34, 35, 36 | prdsplusgval 17518 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎(+g‘𝑌)𝑏) = (𝑐 ∈ 𝐼 ↦ ((𝑎‘𝑐)(+g‘(𝑅‘𝑐))(𝑏‘𝑐)))) |
| 38 | 3, 14, 16, 19, 22, 35, 34, 36 | prdsplusgval 17518 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑏(+g‘𝑌)𝑎) = (𝑐 ∈ 𝐼 ↦ ((𝑏‘𝑐)(+g‘(𝑅‘𝑐))(𝑎‘𝑐)))) |
| 39 | 33, 37, 38 | 3eqtr4d 2787 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎(+g‘𝑌)𝑏) = (𝑏(+g‘𝑌)𝑎)) |
| 40 | 1, 2, 11, 39 | iscmnd 19812 | 1 ⊢ (𝜑 → 𝑌 ∈ CMnd) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ↦ cmpt 5225 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Xscprds 17490 Mndcmnd 18747 CMndccmn 19798 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-prds 17492 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-cmn 19800 |
| This theorem is referenced by: prdsabld 19880 pwscmn 19881 prdsgsum 19999 prdscrngd 20319 |
| Copyright terms: Public domain | W3C validator |