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Mirrors > Home > MPE Home > Th. List > imasaddf | Structured version Visualization version GIF version |
Description: The image structure's group operation is closed in the base set. (Contributed by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
imasaddf.f | ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
imasaddf.e | ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) |
imasaddf.u | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imasaddf.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imasaddf.r | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
imasaddf.p | ⊢ · = (+g‘𝑅) |
imasaddf.a | ⊢ ∙ = (+g‘𝑈) |
imasaddf.c | ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) |
Ref | Expression |
---|---|
imasaddf | ⊢ (𝜑 → ∙ :(𝐵 × 𝐵)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasaddf.f | . 2 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) | |
2 | imasaddf.e | . 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) | |
3 | imasaddf.u | . . 3 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
4 | imasaddf.v | . . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
5 | imasaddf.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
6 | imasaddf.p | . . 3 ⊢ · = (+g‘𝑅) | |
7 | imasaddf.a | . . 3 ⊢ ∙ = (+g‘𝑈) | |
8 | 3, 4, 1, 5, 6, 7 | imasplusg 16607 | . 2 ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
9 | imasaddf.c | . 2 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) | |
10 | 1, 2, 8, 9 | imasaddflem 16620 | 1 ⊢ (𝜑 → ∙ :(𝐵 × 𝐵)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1078 = wceq 1520 ∈ wcel 2079 × cxp 5433 ⟶wf 6213 –onto→wfo 6215 ‘cfv 6217 (class class class)co 7007 Basecbs 16300 +gcplusg 16382 “s cimas 16594 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-sup 8742 df-inf 8743 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-z 11819 df-dec 11937 df-uz 12083 df-fz 12732 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-plusg 16395 df-mulr 16396 df-sca 16398 df-vsca 16399 df-ip 16400 df-tset 16401 df-ple 16402 df-ds 16404 df-imas 16598 |
This theorem is referenced by: imasmnd2 17754 imasgrp2 17959 |
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