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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 6919 | . . 3 ⊢ (seq𝑀( + , 𝐹)‘𝑁) ∈ V | |
| 2 | fvi 6985 | . . 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 7464 | . . . . 5 ⊢ (𝑥 + 𝑦) ∈ V | |
| 9 | fvi 6985 | . . . . 5 ⊢ ((𝑥 + 𝑦) ∈ V → ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦)) | |
| 10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ( I ‘(𝑥 + 𝑦)) = (𝑥 + 𝑦) |
| 11 | fvi 6985 | . . . . . 6 ⊢ (𝑥 ∈ V → ( I ‘𝑥) = 𝑥) | |
| 12 | 11 | elv 3485 | . . . . 5 ⊢ ( I ‘𝑥) = 𝑥 |
| 13 | fvi 6985 | . . . . . 6 ⊢ (𝑦 ∈ V → ( I ‘𝑦) = 𝑦) | |
| 14 | 13 | elv 3485 | . . . . 5 ⊢ ( I ‘𝑦) = 𝑦 |
| 15 | 12, 14 | oveq12i 7443 | . . . 4 ⊢ (( I ‘𝑥)𝑄( I ‘𝑦)) = (𝑥𝑄𝑦) |
| 16 | 7, 10, 15 | 3eqtr4g 2802 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ( I ‘(𝑥 + 𝑦)) = (( I ‘𝑥)𝑄( I ‘𝑦))) |
| 17 | fvex 6919 | . . . 4 ⊢ (𝐹‘𝑥) ∈ V | |
| 18 | fvi 6985 | . . . 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 14090 | . 2 ⊢ (𝜑 → ( I ‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐹)‘𝑁)) |
| 21 | 3, 20 | eqtr3id 2791 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀(𝑄, 𝐹)‘𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 I cid 5577 ‘cfv 6561 (class class class)co 7431 ℤ≥cuz 12878 ...cfz 13547 seqcseq 14042 |
| 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-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-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-iun 4993 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-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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 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-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043 |
| This theorem is referenced by: seqfeq3 14093 gsumpropd2lem 18692 gsumzoppg 19962 |
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