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Mirrors > Home > MPE Home > Th. List > seqeq2 | Structured version Visualization version GIF version |
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
Ref | Expression |
---|---|
seqeq2 | ⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq 7165 | . . . . . 6 ⊢ ( + = 𝑄 → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦𝑄(𝐹‘(𝑥 + 1)))) | |
2 | 1 | opeq2d 4813 | . . . . 5 ⊢ ( + = 𝑄 → 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉 = 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉) |
3 | 2 | mpoeq3dv 7236 | . . . 4 ⊢ ( + = 𝑄 → (𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉) = (𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉)) |
4 | rdgeq1 8050 | . . . 4 ⊢ ((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉) = (𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉) → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ ( + = 𝑄 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)) |
6 | 5 | imaeq1d 5931 | . 2 ⊢ ( + = 𝑄 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω)) |
7 | df-seq 13373 | . 2 ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
8 | df-seq 13373 | . 2 ⊢ seq𝑀(𝑄, 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | |
9 | 6, 7, 8 | 3eqtr4g 2884 | 1 ⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 Vcvv 3497 〈cop 4576 “ cima 5561 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 ωcom 7583 reccrdg 8048 1c1 10541 + caddc 10543 seqcseq 13372 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-xp 5564 df-cnv 5566 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-iota 6317 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-seq 13373 |
This theorem is referenced by: seqeq2d 13379 sadcom 15815 ressmulgnn 30674 cvmliftlem15 32549 |
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