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Theorem seq3f1oleml 10489
Description: Lemma for seq3f1o 10490. 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 4003 . . . . 5 (𝑎 = 𝑥 → (𝑎𝑁𝑥𝑁))
9 2fveq3 5516 . . . . 5 (𝑎 = 𝑥 → (𝐺‘(𝑓𝑎)) = (𝐺‘(𝑓𝑥)))
108, 9ifbieq1d 3556 . . . 4 (𝑎 = 𝑥 → if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)) = if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
1110cbvmptv 4096 . . 3 (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))) = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
121, 2, 3, 4, 5, 6, 7, 11seq3f1olemp 10488 . 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 5511 . . . . . 6 (𝑏 = 𝑥 → (𝑓𝑏) = (𝑓𝑥))
14 id 19 . . . . . 6 (𝑏 = 𝑥𝑏 = 𝑥)
1513, 14eqeq12d 2192 . . . . 5 (𝑏 = 𝑥 → ((𝑓𝑏) = 𝑏 ↔ (𝑓𝑥) = 𝑥))
1615cbvralv 2703 . . . 4 (∀𝑏 ∈ (𝑀...𝑁)(𝑓𝑏) = 𝑏 ↔ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥)
17163anbi2i 1191 . . 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 1005 . . . 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 276 . . . . 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 10007 . . . . . . . 8 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ𝑀))
2120adantl 277 . . . . . . 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 10014 . . . . . . . . . 10 (𝑘 ∈ (𝑀...𝑁) → 𝑘𝑁)
2322adantl 277 . . . . . . . . 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 3541 . . . . . . . 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 5511 . . . . . . . . . . . 12 (𝑏 = 𝑘 → (𝑓𝑏) = (𝑓𝑘))
26 id 19 . . . . . . . . . . . 12 (𝑏 = 𝑘𝑏 = 𝑘)
2725, 26eqeq12d 2192 . . . . . . . . . . 11 (𝑏 = 𝑘 → ((𝑓𝑏) = 𝑏 ↔ (𝑓𝑘) = 𝑘))
28 simplr2 1040 . . . . . . . . . . 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 110 . . . . . . . . . . 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 2846 . . . . . . . . . 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 5515 . . . . . . . . 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 5511 . . . . . . . . . . 11 (𝑥 = 𝑘 → (𝐺𝑥) = (𝐺𝑘))
3332eleq1d 2246 . . . . . . . . . 10 (𝑥 = 𝑘 → ((𝐺𝑥) ∈ 𝑆 ↔ (𝐺𝑘) ∈ 𝑆))
346ralrimiva 2550 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ (ℤ𝑀)(𝐺𝑥) ∈ 𝑆)
3534ad2antrr 488 . . . . . . . . . 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 2846 . . . . . . . . 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 2254 . . . . . . . 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 2254 . . . . . . 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 4003 . . . . . . . . 9 (𝑎 = 𝑘 → (𝑎𝑁𝑘𝑁))
40 2fveq3 5516 . . . . . . . . 9 (𝑎 = 𝑘 → (𝐺‘(𝑓𝑎)) = (𝐺‘(𝑓𝑘)))
4139, 40ifbieq1d 3556 . . . . . . . 8 (𝑎 = 𝑘 → if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)) = if(𝑘𝑁, (𝐺‘(𝑓𝑘)), (𝐺𝑀)))
42 eqid 2177 . . . . . . . 8 (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀))) = (𝑎 ∈ (ℤ𝑀) ↦ if(𝑎𝑁, (𝐺‘(𝑓𝑎)), (𝐺𝑀)))
4341, 42fvmptg 5588 . . . . . . 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 411 . . . . . 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 2214 . . . . 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 110 . . . . . . 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 5511 . . . . . . . . . 10 (𝑎 = (𝑓𝑥) → (𝐺𝑎) = (𝐺‘(𝑓𝑥)))
4847eleq1d 2246 . . . . . . . . 9 (𝑎 = (𝑓𝑥) → ((𝐺𝑎) ∈ 𝑆 ↔ (𝐺‘(𝑓𝑥)) ∈ 𝑆))
4934ad3antrrr 492 . . . . . . . . . 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 5511 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝐺𝑎) = (𝐺𝑥))
5150eleq1d 2246 . . . . . . . . . . 11 (𝑎 = 𝑥 → ((𝐺𝑎) ∈ 𝑆 ↔ (𝐺𝑥) ∈ 𝑆))
5251cbvralv 2703 . . . . . . . . . 10 (∀𝑎 ∈ (ℤ𝑀)(𝐺𝑎) ∈ 𝑆 ↔ ∀𝑥 ∈ (ℤ𝑀)(𝐺𝑥) ∈ 𝑆)
5349, 52sylibr 134 . . . . . . . . 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 1003 . . . . . . . . . . . . 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 488 . . . . . . . . . . . 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 5457 . . . . . . . . . . . 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 110 . . . . . . . . . . . 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 276 . . . . . . . . . . . . 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 9526 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
614, 60syl 14 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℤ)
6261ad3antrrr 492 . . . . . . . . . . . . 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 10003 . . . . . . . . . . . . 13 ((𝑥 ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑥𝑁))
6459, 62, 63syl2anc 411 . . . . . . . . . . . 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 167 . . . . . . . . . . 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, 65ffvelcdmd 5648 . . . . . . . . . 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 10007 . . . . . . . . . 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 2846 . . . . . . . 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 5511 . . . . . . . . . 10 (𝑎 = 𝑀 → (𝐺𝑎) = (𝐺𝑀))
7170eleq1d 2246 . . . . . . . . 9 (𝑎 = 𝑀 → ((𝐺𝑎) ∈ 𝑆 ↔ (𝐺𝑀) ∈ 𝑆))
7234, 52sylibr 134 . . . . . . . . . 10 (𝜑 → ∀𝑎 ∈ (ℤ𝑀)(𝐺𝑎) ∈ 𝑆)
7372ad3antrrr 492 . . . . . . . . 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 9522 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
754, 74syl 14 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
7675ad3antrrr 492 . . . . . . . . . 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 9531 . . . . . . . . . 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 2846 . . . . . . . 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 9526 . . . . . . . . . 10 (𝑥 ∈ (ℤ𝑀) → 𝑥 ∈ ℤ)
8180adantl 277 . . . . . . . . 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 488 . . . . . . . . 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 9318 . . . . . . . . 9 ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑥𝑁)
8481, 82, 83syl2anc 411 . . . . . . . 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 3563 . . . . . . 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 5588 . . . . . . 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 411 . . . . . 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 2254 . . . . 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 477 . . . . 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 477 . . . . 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 10457 . . . 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 2212 . . 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 288 . 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 1898 1 (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  DECID wdc 834  w3a 978   = wceq 1353  wcel 2148  wral 2455  ifcif 3534   class class class wbr 4000  cmpt 4061  wf 5208  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  cle 7983  cz 9242  cuz 9517  ...cfz 9995  seqcseq 10431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-1o 6411  df-er 6529  df-en 6735  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996  df-fzo 10129  df-seqfrec 10432
This theorem is referenced by:  seq3f1o  10490
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