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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumzresunsn | Structured version Visualization version GIF version |
Description: Append an element to a finite group sum expressed as a function restriction. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
Ref | Expression |
---|---|
gsumzresunsn.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumzresunsn.p | ⊢ + = (+g‘𝐺) |
gsumzresunsn.z | ⊢ 𝑍 = (Cntz‘𝐺) |
gsumzresunsn.y | ⊢ 𝑌 = (𝐹‘𝑋) |
gsumzresunsn.f | ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
gsumzresunsn.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
gsumzresunsn.g | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
gsumzresunsn.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
gsumzresunsn.2 | ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐴) |
gsumzresunsn.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐶) |
gsumzresunsn.4 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
gsumzresunsn.5 | ⊢ (𝜑 → (𝐹 “ (𝐴 ∪ {𝑋})) ⊆ (𝑍‘(𝐹 “ (𝐴 ∪ {𝑋})))) |
Ref | Expression |
---|---|
gsumzresunsn | ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ∪ {𝑋}))) = ((𝐺 Σg (𝐹 ↾ 𝐴)) + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumzresunsn.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumzresunsn.p | . . 3 ⊢ + = (+g‘𝐺) | |
3 | gsumzresunsn.z | . . 3 ⊢ 𝑍 = (Cntz‘𝐺) | |
4 | eqid 2735 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)) = (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)) | |
5 | gsumzresunsn.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
6 | gsumzresunsn.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
7 | gsumzresunsn.5 | . . . 4 ⊢ (𝜑 → (𝐹 “ (𝐴 ∪ {𝑋})) ⊆ (𝑍‘(𝐹 “ (𝐴 ∪ {𝑋})))) | |
8 | df-ima 5702 | . . . . 5 ⊢ (𝐹 “ (𝐴 ∪ {𝑋})) = ran (𝐹 ↾ (𝐴 ∪ {𝑋})) | |
9 | gsumzresunsn.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) | |
10 | gsumzresunsn.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
11 | gsumzresunsn.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐶) | |
12 | 11 | snssd 4814 | . . . . . . . 8 ⊢ (𝜑 → {𝑋} ⊆ 𝐶) |
13 | 10, 12 | unssd 4202 | . . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ {𝑋}) ⊆ 𝐶) |
14 | 9, 13 | feqresmpt 6978 | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∪ {𝑋})) = (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) |
15 | 14 | rneqd 5952 | . . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (𝐴 ∪ {𝑋})) = ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) |
16 | 8, 15 | eqtrid 2787 | . . . 4 ⊢ (𝜑 → (𝐹 “ (𝐴 ∪ {𝑋})) = ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) |
17 | 16 | fveq2d 6911 | . . . 4 ⊢ (𝜑 → (𝑍‘(𝐹 “ (𝐴 ∪ {𝑋}))) = (𝑍‘ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)))) |
18 | 7, 16, 17 | 3sstr3d 4042 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)) ⊆ (𝑍‘ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)))) |
19 | 9 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐶⟶𝐵) |
20 | 10 | sselda 3995 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐶) |
21 | 19, 20 | ffvelcdmd 7105 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
22 | gsumzresunsn.2 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐴) | |
23 | gsumzresunsn.4 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
24 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
25 | 24 | fveq2d 6911 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋)) |
26 | gsumzresunsn.y | . . . 4 ⊢ 𝑌 = (𝐹‘𝑋) | |
27 | 25, 26 | eqtr4di 2793 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = 𝑌) |
28 | 1, 2, 3, 4, 5, 6, 18, 21, 11, 22, 23, 27 | gsumzunsnd 19989 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) = ((𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) + 𝑌)) |
29 | 14 | oveq2d 7447 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ∪ {𝑋}))) = (𝐺 Σg (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)))) |
30 | 9, 10 | feqresmpt 6978 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
31 | 30 | oveq2d 7447 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐴)) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))) |
32 | 31 | oveq1d 7446 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐴)) + 𝑌) = ((𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) + 𝑌)) |
33 | 28, 29, 32 | 3eqtr4d 2785 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ∪ {𝑋}))) = ((𝐺 Σg (𝐹 ↾ 𝐴)) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ⊆ wss 3963 {csn 4631 ↦ cmpt 5231 ran crn 5690 ↾ cres 5691 “ cima 5692 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 Basecbs 17245 +gcplusg 17298 Σg cgsu 17487 Mndcmnd 18760 Cntzccntz 19346 |
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-rmo 3378 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-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 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-se 5642 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-isom 6572 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-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-oi 9548 df-card 9977 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-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-gsum 17489 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 |
This theorem is referenced by: rprmdvdsprod 33542 |
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