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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 2818 | . . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
5 | 3, 4 | frlmpws 20822 | . . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌))) |
6 | 1, 2, 5 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌))) |
7 | 6 | fveq2d 6667 | . . . 4 ⊢ (𝜑 → (+g‘𝑌) = (+g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌)))) |
8 | frlmplusgval.p | . . . 4 ⊢ ✚ = (+g‘𝑌) | |
9 | fvex 6676 | . . . . 5 ⊢ (Base‘𝑌) ∈ V | |
10 | eqid 2818 | . . . . . 6 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌)) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌)) | |
11 | eqid 2818 | . . . . . 6 ⊢ (+g‘((ringLMod‘𝑅) ↑s 𝐼)) = (+g‘((ringLMod‘𝑅) ↑s 𝐼)) | |
12 | 10, 11 | ressplusg 16600 | . . . . 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 2878 | . . 3 ⊢ (𝜑 → ✚ = (+g‘((ringLMod‘𝑅) ↑s 𝐼))) |
15 | 14 | oveqd 7162 | . 2 ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹(+g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺)) |
16 | eqid 2818 | . . 3 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
17 | eqid 2818 | . . 3 ⊢ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)) | |
18 | fvexd 6678 | . . 3 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ V) | |
19 | frlmplusgval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
20 | 3, 19 | frlmpws 20822 | . . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
21 | 1, 2, 20 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
22 | 21 | fveq2d 6667 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
23 | 19, 22 | syl5eq 2865 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
24 | eqid 2818 | . . . . . 6 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) | |
25 | 24, 17 | ressbasss 16544 | . . . . 5 ⊢ (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) |
26 | 23, 25 | eqsstrdi 4018 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
27 | frlmplusgval.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
28 | 26, 27 | sseldd 3965 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
29 | frlmplusgval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
30 | 26, 29 | sseldd 3965 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
31 | frlmplusgval.a | . . . 4 ⊢ + = (+g‘𝑅) | |
32 | rlmplusg 19897 | . . . 4 ⊢ (+g‘𝑅) = (+g‘(ringLMod‘𝑅)) | |
33 | 31, 32 | eqtri 2841 | . . 3 ⊢ + = (+g‘(ringLMod‘𝑅)) |
34 | 16, 17, 18, 2, 28, 30, 33, 11 | pwsplusgval 16751 | . 2 ⊢ (𝜑 → (𝐹(+g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺) = (𝐹 ∘f + 𝐺)) |
35 | 15, 34 | eqtrd 2853 | 1 ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘f + 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 Basecbs 16471 ↾s cress 16472 +gcplusg 16553 ↑s cpws 16708 ringLModcrglmod 19870 freeLMod cfrlm 20818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-prds 16709 df-pws 16711 df-sra 19873 df-rgmod 19874 df-dsmm 20804 df-frlm 20819 |
This theorem is referenced by: frlmvplusgvalc 20839 frlmphl 20853 frlmup1 20870 matplusg2 20964 zlmodzxzadd 44334 |
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