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Mirrors > Home > MPE Home > Th. List > seqfeq4 | Structured version Visualization version GIF version |
Description: Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
seqfeq4.m | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
seqfeq4.f | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) |
seqfeq4.cl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
seqfeq4.id | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦)) |
Ref | Expression |
---|---|
seqfeq4 | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀(𝑄, 𝐹)‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6683 | . . 3 ⊢ (seq𝑀( + , 𝐹)‘𝑁) ∈ V | |
2 | fvi 6740 | . . 3 ⊢ ((seq𝑀( + , 𝐹)‘𝑁) ∈ V → ( I ‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ( I ‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁) |
4 | seqfeq4.cl | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
5 | seqfeq4.f | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) | |
6 | seqfeq4.m | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
7 | seqfeq4.id | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦)) | |
8 | ovex 7189 | . . . . 5 ⊢ (𝑥 + 𝑦) ∈ V | |
9 | fvi 6740 | . . . . 5 ⊢ ((𝑥 + 𝑦) ∈ V → ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦)) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦) |
11 | fvi 6740 | . . . . . 6 ⊢ (𝑥 ∈ V → ( I ‘𝑥) = 𝑥) | |
12 | 11 | elv 3499 | . . . . 5 ⊢ ( I ‘𝑥) = 𝑥 |
13 | fvi 6740 | . . . . . 6 ⊢ (𝑦 ∈ V → ( I ‘𝑦) = 𝑦) | |
14 | 13 | elv 3499 | . . . . 5 ⊢ ( I ‘𝑦) = 𝑦 |
15 | 12, 14 | oveq12i 7168 | . . . 4 ⊢ (( I ‘𝑥)𝑄( I ‘𝑦)) = (𝑥𝑄𝑦) |
16 | 7, 10, 15 | 3eqtr4g 2881 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ( I ‘(𝑥 + 𝑦)) = (( I ‘𝑥)𝑄( I ‘𝑦))) |
17 | fvex 6683 | . . . 4 ⊢ (𝐹‘𝑥) ∈ V | |
18 | fvi 6740 | . . . 4 ⊢ ((𝐹‘𝑥) ∈ V → ( I ‘(𝐹‘𝑥)) = (𝐹‘𝑥)) | |
19 | 17, 18 | mp1i 13 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ( I ‘(𝐹‘𝑥)) = (𝐹‘𝑥)) |
20 | 4, 5, 6, 16, 19 | seqhomo 13418 | . 2 ⊢ (𝜑 → ( I ‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐹)‘𝑁)) |
21 | 3, 20 | syl5eqr 2870 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀(𝑄, 𝐹)‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 I cid 5459 ‘cfv 6355 (class class class)co 7156 ℤ≥cuz 12244 ...cfz 12893 seqcseq 13370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-seq 13371 |
This theorem is referenced by: seqfeq3 13421 gsumpropd2lem 17889 gsumzoppg 19064 |
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