| 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 2737 | . . 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 5698 | . . . . 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 4809 | . . . . . . . 8 ⊢ (𝜑 → {𝑋} ⊆ 𝐶) |
| 13 | 10, 12 | unssd 4192 | . . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ {𝑋}) ⊆ 𝐶) |
| 14 | 9, 13 | feqresmpt 6978 | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∪ {𝑋})) = (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) |
| 15 | 14 | rneqd 5949 | . . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (𝐴 ∪ {𝑋})) = ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) |
| 16 | 8, 15 | eqtrid 2789 | . . . 4 ⊢ (𝜑 → (𝐹 “ (𝐴 ∪ {𝑋})) = ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) |
| 17 | 16 | fveq2d 6910 | . . . 4 ⊢ (𝜑 → (𝑍‘(𝐹 “ (𝐴 ∪ {𝑋}))) = (𝑍‘ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)))) |
| 18 | 7, 16, 17 | 3sstr3d 4038 | . . 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 3983 | . . . 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 6910 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋)) |
| 26 | gsumzresunsn.y | . . . 4 ⊢ 𝑌 = (𝐹‘𝑋) | |
| 27 | 25, 26 | eqtr4di 2795 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = 𝑌) |
| 28 | 1, 2, 3, 4, 5, 6, 18, 21, 11, 22, 23, 27 | gsumzunsnd 19974 | . 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 2787 | 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 1540 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 {csn 4626 ↦ cmpt 5225 ran crn 5686 ↾ cres 5687 “ cima 5688 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 Basecbs 17247 +gcplusg 17297 Σg cgsu 17485 Mndcmnd 18747 Cntzccntz 19333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-gsum 17487 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 |
| This theorem is referenced by: rprmdvdsprod 33562 |
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