Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > frlmplusgval | Structured version Visualization version GIF version |
Description: Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
Ref | Expression |
---|---|
frlmplusgval.y | ⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
frlmplusgval.b | ⊢ 𝐵 = (Base‘𝑌) |
frlmplusgval.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
frlmplusgval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
frlmplusgval.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
frlmplusgval.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
frlmplusgval.a | ⊢ + = (+g‘𝑅) |
frlmplusgval.p | ⊢ ✚ = (+g‘𝑌) |
Ref | Expression |
---|---|
frlmplusgval | ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘f + 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmplusgval.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
2 | frlmplusgval.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
3 | frlmplusgval.y | . . . . . . 7 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
4 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
5 | 3, 4 | frlmpws 20888 | . . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌))) |
6 | 1, 2, 5 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌))) |
7 | 6 | fveq2d 6669 | . . . 4 ⊢ (𝜑 → (+g‘𝑌) = (+g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌)))) |
8 | frlmplusgval.p | . . . 4 ⊢ ✚ = (+g‘𝑌) | |
9 | fvex 6678 | . . . . 5 ⊢ (Base‘𝑌) ∈ V | |
10 | eqid 2821 | . . . . . 6 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌)) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌)) | |
11 | eqid 2821 | . . . . . 6 ⊢ (+g‘((ringLMod‘𝑅) ↑s 𝐼)) = (+g‘((ringLMod‘𝑅) ↑s 𝐼)) | |
12 | 10, 11 | ressplusg 16606 | . . . . 5 ⊢ ((Base‘𝑌) ∈ V → (+g‘((ringLMod‘𝑅) ↑s 𝐼)) = (+g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌)))) |
13 | 9, 12 | ax-mp 5 | . . . 4 ⊢ (+g‘((ringLMod‘𝑅) ↑s 𝐼)) = (+g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌))) |
14 | 7, 8, 13 | 3eqtr4g 2881 | . . 3 ⊢ (𝜑 → ✚ = (+g‘((ringLMod‘𝑅) ↑s 𝐼))) |
15 | 14 | oveqd 7167 | . 2 ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹(+g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺)) |
16 | eqid 2821 | . . 3 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
17 | eqid 2821 | . . 3 ⊢ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)) | |
18 | fvexd 6680 | . . 3 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ V) | |
19 | frlmplusgval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
20 | 3, 19 | frlmpws 20888 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
21 | 1, 2, 20 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
22 | 21 | fveq2d 6669 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
23 | 19, 22 | syl5eq 2868 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
24 | eqid 2821 | . . . . . 6 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) | |
25 | 24, 17 | ressbasss 16550 | . . . . 5 ⊢ (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) |
26 | 23, 25 | eqsstrdi 4021 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
27 | frlmplusgval.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
28 | 26, 27 | sseldd 3968 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
29 | frlmplusgval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
30 | 26, 29 | sseldd 3968 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
31 | frlmplusgval.a | . . . 4 ⊢ + = (+g‘𝑅) | |
32 | rlmplusg 19962 | . . . 4 ⊢ (+g‘𝑅) = (+g‘(ringLMod‘𝑅)) | |
33 | 31, 32 | eqtri 2844 | . . 3 ⊢ + = (+g‘(ringLMod‘𝑅)) |
34 | 16, 17, 18, 2, 28, 30, 33, 11 | pwsplusgval 16757 | . 2 ⊢ (𝜑 → (𝐹(+g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺) = (𝐹 ∘f + 𝐺)) |
35 | 15, 34 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘f + 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ‘cfv 6350 (class class class)co 7150 ∘f cof 7401 Basecbs 16477 ↾s cress 16478 +gcplusg 16559 ↑s cpws 16714 ringLModcrglmod 19935 freeLMod cfrlm 20884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-hom 16583 df-cco 16584 df-prds 16715 df-pws 16717 df-sra 19938 df-rgmod 19939 df-dsmm 20870 df-frlm 20885 |
This theorem is referenced by: frlmvplusgvalc 20905 frlmphl 20919 frlmup1 20936 matplusg2 21030 zlmodzxzadd 44399 |
Copyright terms: Public domain | W3C validator |