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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 2735 | . . . . . . 7 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
5 | 3, 4 | frlmpws 21788 | . . . . . 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 6911 | . . . 4 ⊢ (𝜑 → (+g‘𝑌) = (+g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌)))) |
8 | frlmplusgval.p | . . . 4 ⊢ ✚ = (+g‘𝑌) | |
9 | fvex 6920 | . . . . 5 ⊢ (Base‘𝑌) ∈ V | |
10 | eqid 2735 | . . . . . 6 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌)) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝑌)) | |
11 | eqid 2735 | . . . . . 6 ⊢ (+g‘((ringLMod‘𝑅) ↑s 𝐼)) = (+g‘((ringLMod‘𝑅) ↑s 𝐼)) | |
12 | 10, 11 | ressplusg 17336 | . . . . 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 2800 | . . 3 ⊢ (𝜑 → ✚ = (+g‘((ringLMod‘𝑅) ↑s 𝐼))) |
15 | 14 | oveqd 7448 | . 2 ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹(+g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺)) |
16 | eqid 2735 | . . 3 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
17 | eqid 2735 | . . 3 ⊢ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)) | |
18 | fvexd 6922 | . . 3 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ V) | |
19 | frlmplusgval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑌) | |
20 | 3, 19 | frlmpws 21788 | . . . . . . . 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 6911 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
23 | 19, 22 | eqtrid 2787 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
24 | eqid 2735 | . . . . . 6 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) | |
25 | 24, 17 | ressbasss 17284 | . . . . 5 ⊢ (Base‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) |
26 | 23, 25 | eqsstrdi 4050 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
27 | frlmplusgval.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
28 | 26, 27 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
29 | frlmplusgval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
30 | 26, 29 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
31 | frlmplusgval.a | . . . 4 ⊢ + = (+g‘𝑅) | |
32 | rlmplusg 21219 | . . . 4 ⊢ (+g‘𝑅) = (+g‘(ringLMod‘𝑅)) | |
33 | 31, 32 | eqtri 2763 | . . 3 ⊢ + = (+g‘(ringLMod‘𝑅)) |
34 | 16, 17, 18, 2, 28, 30, 33, 11 | pwsplusgval 17537 | . 2 ⊢ (𝜑 → (𝐹(+g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺) = (𝐹 ∘f + 𝐺)) |
35 | 15, 34 | eqtrd 2775 | 1 ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘f + 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 Basecbs 17245 ↾s cress 17274 +gcplusg 17298 ↑s cpws 17493 ringLModcrglmod 21189 freeLMod cfrlm 21784 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-prds 17494 df-pws 17496 df-sra 21190 df-rgmod 21191 df-dsmm 21770 df-frlm 21785 |
This theorem is referenced by: frlmvplusgvalc 21805 frlmphl 21819 frlmup1 21836 matplusg2 22449 zlmodzxzadd 48203 |
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