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Theorem seq3f1oleml 9995
 Description: Lemma for seq3f1o 9996. This is more or less the result, but stated in terms of 𝐹 and 𝐺 without 𝐻. 𝐿 and 𝐻 may differ in terms of what happens to terms after 𝑁. The terms after 𝑁 don't matter for the value at 𝑁 but we need some definition given the way our theorems concerning seq work. (Contributed by Jim Kingdon, 17-Aug-2022.)
Hypotheses
Ref Expression
iseqf1o.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqf1o.2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
iseqf1o.3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
iseqf1o.4 (𝜑𝑁 ∈ (ℤ𝑀))
iseqf1o.6 (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1o.7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
iseqf1o.l 𝐿 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝐹𝑥)), (𝐺𝑀)))
Assertion
Ref Expression
seq3f1oleml (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
Distinct variable groups:   𝑥, + ,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐿,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝑥,𝑁,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem seq3f1oleml
Dummy variables 𝑓 𝑘 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqf1o.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2 iseqf1o.2 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3 iseqf1o.3 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
4 iseqf1o.4 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
5 iseqf1o.6 . . 3 (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
6 iseqf1o.7 . . 3 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
7 iseqf1o.l . . 3 𝐿 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝐹𝑥)), (𝐺𝑀)))
8 breq1 3856 . . . . 5 (𝑎 = 𝑥 → (𝑎𝑁𝑥𝑁))
9 2fveq3 5325 . . . . 5 (𝑎 = 𝑥 → (𝐺‘(𝑓𝑎)) = (𝐺‘(𝑓𝑥)))
108, 9ifbieq1d 3419 . . . 4 (𝑎 = 𝑥 → if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)) = if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
1110cbvmptv 3942 . . 3 (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))) = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
121, 2, 3, 4, 5, 6, 7, 11seq3f1olemp 9994 . 2 (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
13 fveq2 5320 . . . . . 6 (𝑏 = 𝑥 → (𝑓𝑏) = (𝑓𝑥))
14 id 19 . . . . . 6 (𝑏 = 𝑥𝑏 = 𝑥)
1513, 14eqeq12d 2103 . . . . 5 (𝑏 = 𝑥 → ((𝑓𝑏) = 𝑏 ↔ (𝑓𝑥) = 𝑥))
1615cbvralv 2593 . . . 4 (∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ↔ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥)
17163anbi2i 1136 . . 3 ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
18 simpr3 952 . . . 4 ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
194adantr 271 . . . . 5 ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝑁 ∈ (ℤ𝑀))
20 elfzuz 9499 . . . . . . . 8 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ𝑀))
2120adantl 272 . . . . . . 7 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ (ℤ𝑀))
22 elfzle2 9505 . . . . . . . . . 10 (𝑘 ∈ (𝑀...𝑁) → 𝑘𝑁)
2322adantl 272 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘𝑁)
2423iftrued 3406 . . . . . . . 8 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)) = (𝐺‘(𝑓𝑘)))
25 fveq2 5320 . . . . . . . . . . . 12 (𝑏 = 𝑘 → (𝑓𝑏) = (𝑓𝑘))
26 id 19 . . . . . . . . . . . 12 (𝑏 = 𝑘𝑏 = 𝑘)
2725, 26eqeq12d 2103 . . . . . . . . . . 11 (𝑏 = 𝑘 → ((𝑓𝑏) = 𝑏 ↔ (𝑓𝑘) = 𝑘))
28 simplr2 987 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏)
29 simpr 109 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
3027, 28, 29rspcdva 2730 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑓𝑘) = 𝑘)
3130fveq2d 5324 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘(𝑓𝑘)) = (𝐺𝑘))
32 fveq2 5320 . . . . . . . . . . 11 (𝑥 = 𝑘 → (𝐺𝑥) = (𝐺𝑘))
3332eleq1d 2157 . . . . . . . . . 10 (𝑥 = 𝑘 → ((𝐺𝑥) ∈ 𝑆 ↔ (𝐺𝑘) ∈ 𝑆))
346ralrimiva 2447 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐺𝑥) ∈ 𝑆)
3534ad2antrr 473 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ∀𝑥 ∈ (ℤ𝑀)(𝐺𝑥) ∈ 𝑆)
3633, 35, 21rspcdva 2730 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺𝑘) ∈ 𝑆)
3731, 36eqeltrd 2165 . . . . . . . 8 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘(𝑓𝑘)) ∈ 𝑆)
3824, 37eqeltrd 2165 . . . . . . 7 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)) ∈ 𝑆)
39 breq1 3856 . . . . . . . . 9 (𝑎 = 𝑘 → (𝑎𝑁𝑘𝑁))
40 2fveq3 5325 . . . . . . . . 9 (𝑎 = 𝑘 → (𝐺‘(𝑓𝑎)) = (𝐺‘(𝑓𝑘)))
4139, 40ifbieq1d 3419 . . . . . . . 8 (𝑎 = 𝑘 → if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)) = if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)))
42 eqid 2089 . . . . . . . 8 (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))) = (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))
4341, 42fvmptg 5395 . . . . . . 7 ((𝑘 ∈ (ℤ𝑀) ∧ if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)) ∈ 𝑆) → ((𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))‘𝑘) = if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)))
4421, 38, 43syl2anc 404 . . . . . 6 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))‘𝑘) = if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)))
4544, 24, 313eqtrd 2125 . . . . 5 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))‘𝑘) = (𝐺𝑘))
46 simpr 109 . . . . . . 7 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → 𝑥 ∈ (ℤ𝑀))
47 fveq2 5320 . . . . . . . . . 10 (𝑎 = (𝑓𝑥) → (𝐺𝑎) = (𝐺‘(𝑓𝑥)))
4847eleq1d 2157 . . . . . . . . 9 (𝑎 = (𝑓𝑥) → ((𝐺𝑎) ∈ 𝑆 ↔ (𝐺‘(𝑓𝑥)) ∈ 𝑆))
4934ad3antrrr 477 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → ∀𝑥 ∈ (ℤ𝑀)(𝐺𝑥) ∈ 𝑆)
50 fveq2 5320 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝐺𝑎) = (𝐺𝑥))
5150eleq1d 2157 . . . . . . . . . . 11 (𝑎 = 𝑥 → ((𝐺𝑎) ∈ 𝑆 ↔ (𝐺𝑥) ∈ 𝑆))
5251cbvralv 2593 . . . . . . . . . 10 (∀𝑎 ∈ (ℤ𝑀)(𝐺𝑎) ∈ 𝑆 ↔ ∀𝑥 ∈ (ℤ𝑀)(𝐺𝑥) ∈ 𝑆)
5349, 52sylibr 133 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → ∀𝑎 ∈ (ℤ𝑀)(𝐺𝑎) ∈ 𝑆)
54 simpr1 950 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
5554ad2antrr 473 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → 𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
56 f1of 5268 . . . . . . . . . . . 12 (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝑓:(𝑀...𝑁)⟶(𝑀...𝑁))
5755, 56syl 14 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → 𝑓:(𝑀...𝑁)⟶(𝑀...𝑁))
58 simpr 109 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → 𝑥𝑁)
5946adantr 271 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → 𝑥 ∈ (ℤ𝑀))
60 eluzelz 9091 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
614, 60syl 14 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℤ)
6261ad3antrrr 477 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → 𝑁 ∈ ℤ)
63 elfz5 9495 . . . . . . . . . . . . 13 ((𝑥 ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑥𝑁))
6459, 62, 63syl2anc 404 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑥𝑁))
6558, 64mpbird 166 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → 𝑥 ∈ (𝑀...𝑁))
6657, 65ffvelrnd 5451 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → (𝑓𝑥) ∈ (𝑀...𝑁))
67 elfzuz 9499 . . . . . . . . . 10 ((𝑓𝑥) ∈ (𝑀...𝑁) → (𝑓𝑥) ∈ (ℤ𝑀))
6866, 67syl 14 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → (𝑓𝑥) ∈ (ℤ𝑀))
6948, 53, 68rspcdva 2730 . . . . . . . 8 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ 𝑥𝑁) → (𝐺‘(𝑓𝑥)) ∈ 𝑆)
70 fveq2 5320 . . . . . . . . . 10 (𝑎 = 𝑀 → (𝐺𝑎) = (𝐺𝑀))
7170eleq1d 2157 . . . . . . . . 9 (𝑎 = 𝑀 → ((𝐺𝑎) ∈ 𝑆 ↔ (𝐺𝑀) ∈ 𝑆))
7234, 52sylibr 133 . . . . . . . . . 10 (𝜑 → ∀𝑎 ∈ (ℤ𝑀)(𝐺𝑎) ∈ 𝑆)
7372ad3antrrr 477 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ ¬ 𝑥𝑁) → ∀𝑎 ∈ (ℤ𝑀)(𝐺𝑎) ∈ 𝑆)
74 eluzel2 9087 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
754, 74syl 14 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
7675ad3antrrr 477 . . . . . . . . . 10 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ ¬ 𝑥𝑁) → 𝑀 ∈ ℤ)
77 uzid 9096 . . . . . . . . . 10 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
7876, 77syl 14 . . . . . . . . 9 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ ¬ 𝑥𝑁) → 𝑀 ∈ (ℤ𝑀))
7971, 73, 78rspcdva 2730 . . . . . . . 8 ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) ∧ ¬ 𝑥𝑁) → (𝐺𝑀) ∈ 𝑆)
80 eluzelz 9091 . . . . . . . . . 10 (𝑥 ∈ (ℤ𝑀) → 𝑥 ∈ ℤ)
8180adantl 272 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → 𝑥 ∈ ℤ)
8261ad2antrr 473 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → 𝑁 ∈ ℤ)
83 zdcle 8886 . . . . . . . . 9 ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑥𝑁)
8481, 82, 83syl2anc 404 . . . . . . . 8 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → DECID 𝑥𝑁)
8569, 79, 84ifcldadc 3426 . . . . . . 7 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)) ∈ 𝑆)
8610, 42fvmptg 5395 . . . . . . 7 ((𝑥 ∈ (ℤ𝑀) ∧ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)) ∈ 𝑆) → ((𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))‘𝑥) = if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
8746, 85, 86syl2anc 404 . . . . . 6 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → ((𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))‘𝑥) = if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
8887, 85eqeltrd 2165 . . . . 5 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → ((𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))‘𝑥) ∈ 𝑆)
896adantlr 462 . . . . 5 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
901adantlr 462 . . . . 5 (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
9119, 45, 88, 89, 90seq3fveq 9958 . . . 4 ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
9218, 91eqtr3d 2123 . . 3 ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
9317, 92sylan2br 283 . 2 ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
9412, 93exlimddv 1827 1 (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 781   ∧ w3a 925   = wceq 1290   ∈ wcel 1439  ∀wral 2360  ifcif 3399   class class class wbr 3853   ↦ cmpt 3907  ⟶wf 5026  –1-1-onto→wf1o 5029  ‘cfv 5030  (class class class)co 5668   ≤ cle 7586  ℤcz 8813  ℤ≥cuz 9082  ...cfz 9487  seqcseq 9915 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-addcom 7508  ax-addass 7510  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-0id 7516  ax-rnegex 7517  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524 This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-if 3400  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-iord 4204  df-on 4206  df-ilim 4207  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-frec 6172  df-1o 6197  df-er 6308  df-en 6514  df-fin 6516  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-inn 8486  df-n0 8737  df-z 8814  df-uz 9083  df-fz 9488  df-fzo 9617  df-iseq 9916  df-seq3 9917 This theorem is referenced by:  seq3f1o  9996
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